The nine days of Navaratri is auspicious for doing Devi worship. Legend says that Devis reside in each houses at this time. If we pray to Lord Vishnu or Lord Brahma or Lord Shiva will the prayer yield results? Puranic reference will be of much helpful to all rather than comments.
There are Nithya Karmas and Naimittika Karmas. The Navaratri celebration falls under Naimittika Tithi or Karma. Whoever deity is your Kula Devata or Ishta Devata for them Nitya Puja is to be done even in Navaratri days also. Along with it, Devi worship is also done.
Devi Bhagavatam says in its Third Skandham Chapter 26 as
नवरात्रे तु संप्राप्ते किं कर्तव्यं द्विजोत्तम । विधानं विधिवद् ब्रूहि शरत्काले विशेषतः ॥
When Navaratri approaches what is the duty, O Dvijottama. Tell me the procedure to be followed especially in Śaratkāla.
किं फलं खलु कस्तत्र विधिः कार्यो महामते । एतद्विस्तरतो ब्रूहि कृपया द्विजसत्तम ॥
What is result? What is to be done then? Please explain them elaborately, O dvijasattama.
व्यास उवाच -
श्रुणु राजन्प्रवक्ष्यामि नवरात्र व्रतं शुभम् । शरत्काले विशेषेण कर्तव्यं विधिपूर्वकम् ॥
Vyasa said -
Listen O Raja, about Navaratri Vrata to be done in Śaratkāla which is auspicious and the procedure to do it.
वसंते च प्रकर्तव्यं तथेव प्रेमपूर्वकम् । द्वावृतू यमदंष्ट्राख्यौ नूनं सर्वजनेषु वै ॥
Vasanta and Śaratkala, both the seasons are like teeth of Yama (Yamadaṃṣṭrā) for all beings.
शरद्वसंतनामानौ दुर्गमौ प्रणिनामिह । तस्माद्यत्नादिदं कार्यं सर्वत्र शुभमिच्छता ॥
द्वावेव सुमहाघोरावृतू रोगकरौ नृणाम् । वसंतशरदावेव सर्वनाशकरावुभौ ॥
Sarat and vasanta are very difficult to cross for beings. So everyone should do Puja for the good of everyone. These two seasons cause many diseases causing lot of destruction.
तस्मात्तत्र प्रकर्तव्यं चंडिकापूजनं बुधैः । चैत्राश्विने शुभे मासे भक्तिपूर्वं नराधिप ॥
So it is the duty of wise people to do Devi (or Chandi) Puja in the months of Chaitra and Āswayuja months with bhakti.
In the same chapter Vyāsa says
प्रातर्नित्यं पुरः कृत्वा द्विजानां वरणं ततः ।
In the early morning (प्रातः), Nitya Karmas (नित्यं) are done (कृत्वा) well in advance (पुरः).
The Chapter 27 of the Third Skanda says
अशक्तो नियतां पूजां कर्तुं चेन्नवरात्रके । अष्टम्यां च विशेषेण कर्तव्यं पूजनं सदा ॥
Whoever is unable to perform Puja following all the prescribed rules, should do Puja on Ashtami day.
उपवासो ह्यशक्तानां नवरात्रव्रते पुनः । उपोषणत्रयं प्रोक्तं यथोक्तफलदं नृप ॥
Those who are unable to fast for Navaratri (all days) can do it for three days to get the same result.
सप्तम्यां च तथाऽष्टम्यां नवम्यां भक्तिभावतः । त्रिरात्रकरणात्सर्वं फलं भवति पूजनात् ॥
If one worships with Bhakti on Saptami, Ashtami and Navami, one obtains full benefit of vrata.
पूजाभिश्चैव होमैश्च कुमारीपूजनैस्तथा । संपूर्णं तद्व्रतं प्रोक्तं तद्विप्राणां चैव भोजनैः ॥
Pūja, Homa, Kumārī Pūja, Brāhmaṇa samārādhana are the essentials for Navaratri Vrata to be complete.
So one can worship other gods too. In addition to those, one should perform Navaratri Vrata also.
Hope this helps you.
Yes, you definitely can worship lord Vishnu and lord Shiva during the time of Navaratri.
These nine days are meant to be auspicious in the Hindu Vedic Culture. Whatever shloka/mantra you recite, or worship of any deity which is done in these nine days is told to be much beneficial.
Apart from this you can definitely worship lord shiva/vishnu/brahma. Vishnu/Shiva being the husbands of Lakshmi/Sati would always fulfill your desires and if you welcome them to your home the wives(Lakshmi/Sati) would definitely accompany their husbands and will fulfill your desires.
Worship of any lord, anytime, performed during these nine days is fruitful.
Navaratri means nine nights. Darkness is associated with night. What is this darkness in a human being? It is the darkness of ignorance. The purpose of the Navaratri celebrations is to enable you to get rid of darkness. When a reference is made to Devi, it signifies the unified form of Durga, Lakshmi and Saraswathi. These three Goddesses together represent energy or Shakti. Shakti is the energy that represents Nature (Prakruti). Nature is made up of the three qualities,Satwa, Rajas and Tamas. Saraswathi represents Satwa guna (purity). Lakshmi represents Rajo Guna (energy and activity) and Durga represents Thamo Guna (dullness or inertia). As Nature is made up of these three qualities (Rajas, Tamas and Satwa), to gain control over one’s inner Nature, people worship Durga, Lakshmi and Saraswathi as they represent these three qualities. To secure the grace of Goddess Shakthi, you must have triple purity - purity of the heart, purity in speech and purity in action.
Source---HH Chandrasekarendra saraswathi swamigal