I have read about how we should develop Sattvic guna to be devotee of Sri Vishnu or His Avatars. But can anyone with rajasic or tamasic guna predominant be devotee of Sri Vishnu? If yes, I would love to hear about such personalities from scriptures.
I. Can one with predominance of Rajas and Tamas worship Lord Vishnu?
Yes. Those with Rajasic and Tamasic Guna predominance can become devotees of Vishnu. Where Vishnu is considered the Supreme, though in his Saguna form as Vishnu he is sattvic, yet as the Supreme God he is the Lord of Maya and above these gunas which are in fact a product of Maya. ŚB 6.1. In the Srimad Bhagavadgita, Lord Krishna himself says:
The same am I to all beings; to Me there is none hateful or dear; but those who worship Me with devotion are in Me and I am also in them. 9.29
Even if the most sinful worships Me, with devotion to none else, he too should indeed by regarded as righteous for he has rightly resolved. 9.30
Adi Shankaracharya’s Bhashya to BG 4.13 states that Kshatriyas, Vaishyas and Shudras variously have either predominance of Rajoguna or Tamoguna and in continuation of the above Lord Krishna says:
For, taking refuge in Me, they also who, O Arjuna, may be of a sinful birth women, Vaisyas as well as Sudras attain the Supreme Goal.
Thus it is very clear that people of all gunas can worship Vishnu.
II. Examples of devotees of Vishnu with Rajasic and Tamasic qualities
Two of the best examples of such people are Valmiki and Ajamila. Then from the Vaishnava perspective even Brahma and Shiva can be considered.
- Valmiki in his previous birth was a hunter who robs a Brahmin named Shankha of his possessions including his sandals. Later on, seeing the Brahmin distressed, the hunter gives him his sandals back, to which the Brahmin remarks:
Indeed this (hunter) is one whose merit has become mature, because he made this gift in the monṃ of Vaiśākha. Viṣṇu becomes delighted even with a hunter of wicked nature. SkP II.7.17
The hunter asks the Brahmin in the subsequent chapter: What was that Karma as a result of which (I got) this birth as a hunter full of Tamoguṇa?
This hunter is then reborn as a Brahmin named Agnisharman who again takes to robbery in his current birth. Here too he is given Shri Rama’s mantra japa by the Saptarishis and is transformed into the great sage Valmiki. SkP V.1.24
Note: the story of Valmiki may contain some inconsistencies within the Skanda Purana itself but the above version is based on the sequence as described by Gita Press Gorakhpur in the preface to the Valmiki Ramayana.
- Ajamila though a brahmana became sinful and of Tamasic nature. However even while dying though he did not remember Lord Vishnu, he still called out his name (actually calling out to one of his sons) and this Tamasic “devotee” was given moksha by the Lord. His story is described in the Srimad Bhagavata Purana 6th as Skandha, Chapter 1:
Ajāmila was a resident of Kanyakubja. He was trained by his parents to become a perfect brāhmaṇa by studying the Vedas and following the regulative principles, but because of his past, this youthful brāhmaṇa was somehow attracted by a prostitute, and because of her association he became most fallen and abandoned all regulative principles. Ajāmila begot in the womb of the prostitute ten sons, the last of whom was called Nārāyaṇa. At the time of Ajāmila’s death, when the order carriers of Yamarāja came to take him, he loudly called the name Nārāyaṇa in fear because he was attached to his youngest son. Thus he remembered the original Nārāyaṇa, Lord Viṣṇu. Although he did not chant the holy name of Nārāyaṇa completely offenselessly, it acted nevertheless. As soon as he chanted the holy name of Nārāyaṇa, the order carriers of Lord Viṣṇu immediately appeared on the scene. A discussion ensued between the order carriers of Lord Viṣṇu and those of Yamarāja, and by hearing that discussion Ajāmila was liberated.
- Since you have used the Vaishnava tag, another example from the Vaishnava perspective would be Lord Brahma and Lord Shiva. As I discuss in this answer, the Trimurti are the creative, preservative and destructive manifestations of the same supreme god and there is no difference between them. They have mutual adoration and devotion for each other. Each of the Trimurti (based on the functions they perform) has one of the three gunas predominant: 1. Brahma has Rajoguna, Vishnu has Satoguna and Shiva has Tamoguna (Ref. In the course of their mutual devotion, there are also several examples where for the benefit of the world Brahma (Rajoguna) and Shiva (Tamoguna) are described as offering their worship to Lord Vishnu, one being the time to take Ramavatara.
When the greatest of gods having predominance of Rajas and Tamas are devotees of Lord Vishnu, and he himself gives affirmation of his acceptance of all in the Bhagavad Gita, then what’s preventing anyone from worshipping the impartial Mayapati?
There's a verse in Srimad Bhagavatam that explains this directly.
tadā rajas-tamo-bhāvāḥ kāma-lobhādayaś ca ye ceta etair anāviddhaṁ sthitaṁ sattve prasīdati
Translation: As soon as irrevocable loving service is established in the heart, the effects of nature's modes of passion and ignorance, such as lust, desire and hankering, disappear from the heart. Then the devotee is established in goodness, and he becomes completely happy.
In regard to personalities, Ajamila (6th canto Srimad Bhagavatam), Nalakuvara, and Manigriva (10th canto Srimad Bhagavatam) are some of the prominent examples.
Ajamila: He began life as a fine and dutiful brāhmaṇa, but in his young manhood became wholly corrupted by a prostitute. At the end of his wicked life, just by calling the name "Nārāyaṇa (Kṛṣṇa)," he was saved despite so much sin. Śukadeva points out that austerity, charity and the performance of ritualistic ceremonies for counteracting sinful activities are recommended processes, but that by performing them one cannot remove the sinful desire-seed from the heart, as was the case with Ajāmila in his youth. This sinful desire-seed can be removed only by achieving Kṛṣṇa consciousness. And this can be accomplished very easily by chanting the mahā-mantra, or Hare Kṛṣṇa mantra, as recommended by Śrī Caitanya Mahāprabhu. In other words, unless one adopts the path of devotional service, he cannot be one-hundred-percent clean from all the reactions of sinful activities. Ref: Nectar of Devotion, chapter 1
Nalakuvara and Manigriva: These two sons of Kuvera (the treasurer of the demigods) were puffed up on account of the opulence of their father, and so once on a heavenly planet they were enjoying themselves in a lake with some naked damsels of heaven. At that time the great saint Nārada Muni was passing on the road and was sorry to see the behavior of the sons of Kuvera. Seeing Nārada passing by, the damsels of heaven covered their bodies with cloth, but the two sons, being drunkards, did not have this decency. Nārada became angry with their behavior and cursed them thus: "You have no sense, so it is better if you become trees instead of the sons of Kuvera." Upon hearing this, the boys came to their senses and begged Nārada to be pardoned for their offenses. Nārada then said, "Yes, you shall become trees, arjuna trees, and you will stand in the courtyard of Nanda Mahārāja. But Kṛṣṇa Himself will appear in time as the foster son of Nanda, and He will deliver you." In other words, the curse of Nārada was a benediction to the sons of Kuvera because indirectly it was foretold that they would be able to receive the favor of Lord Kṛṣṇa. After that, Kuvera's two sons stood as two big arjuna trees in the courtyard of Nanda Mahārāja until Lord Dāmodara, in order to fulfill the desire of Nārada, dragged the grinding mortar to which He was tied and struck the two trees, violently causing them to fall down. From out of these fallen trees came Nalakūvara and Maṇigrīva, who had by then become great devotees of the Lord. Ref: Nectar of Devotion, chapter 4
This proves that the only qualification needed is the eagerness to render service to Supreme Lord Krishna, no matter whichever material modes we are entrapped in. This is stated in Srimad Bhagavatam 1.2.16
The eagerness to hear about God is the first qualification of a devotee eligible for entering the kingdom of God.