Ṛṣi Vyāghrapāda Vāsiṣṭha was the mantradraṣṭā of tṛcā (9.97.16-18) in Ṛg Veda saṁhitā, as per the Anukramaṇī. His gotra was Vasiṣṭha and pravara was ekārṣeya (i.e. Vāsiṣṭha), as mentioned in Matsya Purāṇa (adhyāya 200) and Puruṣottama Paṇḍita's Gotrapravaramañjarī.
Ṛṣi Vyāghrapāda was also the father of Ṛṣi Upamanyu Vāsiṣṭha (Śiva Purāṇa 3.32.2, etc.).
He is usually referred to with reverence in śāstras, signifying he was a really venerable Ṛṣi.
Now, we would talk about the legendary story of Ṛṣi Vyāghrapāda, mentioned by OP. The sthala purāṇa of the Tillai Naṭarāja mandira at Cidambaram is associated with Vyāghrapāda and Patañjali. From the Cidāmara Māhātmyam, Kōyiṟpurāṇam (கோயிற்புராணம்), and Kuñcitāṅghristava, we have details of Vyāghrapāda and Patañjali, who are associated with the history of Cidambaram and who are said to have attained mukti there. Śiva performed his ānandatāṇḍava to bestow divine grace on Vyāghrapāda & Patañjali and the devatās, in the presence of His consort, Śivakāmasundarī.
The book Tillai and Natarajan (pg. 8-9) by B. Natarajan, briefly mentions the story of Vyāghrapāda, based on the aforementioned śāstras.
Vyāghrapāda was the son of Madhyandina Muni, who taught him all the Veda, śāstras and pañcākṣara mantra, according to the Śaivāgamas, and then inquired of him what else he wanted to learn. The son asked his father to instruct him in the best mode of tapa. Taking the father's instructions, the son selected a sacred spot of Tillai for his tapa. He built for himself an āśrama on the west side of Tillai near another tank and did pratiṣṭha of another liṅga there and did pūjā of both the liṅgas. He found that the flowers were spoiled by honey bees when gathered after dawn, and so he prayed to Bhagavān that he might be provided with the eyes, claws, and feet of a tiger to fulfill his desire of collecting flawless flowers for pūjā. His request was granted. Hence his name Vyāghrapāda. After he had spent some time in bhakti, his father got him married, and in due course, a son was born to him, Upamanyu by name. The child was fed by Arundhatī on the milk of divya cow Kāmadhenu. Even as the child reached Vyāghrapāda's home, he wanted the divya milk. Bhagavān wrought a miracle and the oceal of milk of heaven (Tirupparkkadal) was brought to earth to feed the child of the Ṛṣi. One day the Ṛṣi, in a moment of yogic trance, had a vision of the Bhagavān's tāṇḍava before the 48000 munis in Dāruka forest and longed to see it himself; and so awaited the great day when Bhagavān would exhibit his ānandatāṇḍava to him and other bhaktas at Tillai. Since Vyāghrapāda (Pulikkāl muṉivar in Tamil) devoted himself to the Lord of Tillai, the holy centre came to be known Perumparrapuliyur or Puliyor in short (whose praise Appar sang). The Śivaliṅga pūjita by Vyāghrapāda was Sri Mulanathar, the sacred tank was Śivagaṅgā to the north of the Sri Mulanathar temple.
The banyan tree has been symbolised in a stone sculpture with the label "Karpaka Vṛkṣa" on the western prakara of the present Sri Mulanatha shrine. (Some scholars hold the view that this stone sculpture represents the Tillai tree). According to Cidambara māhātmyam, Śiva confirms that on the southern bank of the holy Śivagaṅgā tank, he performs the Ānandatāṇḍava, and Mahādeva as Svayambhuvaliṅga in the Mulasthana shrine on the southern bank of the Śivagaṅgā tank dispels pāpas & ensures mukti from saṁsāra. On the western side of the great Cidambaram mandira complex is mandira of Tiruppulisvaram with Vyāghrapāda tīrtha attached to this temple, whose pratiṣṭhā was done by Vyāghrapāda for pūjāna. It's known more popularly as Ilamayakkinar temple (connected with the episode of Tirunilakantha Nayanar).
