Yes. Long before Aruḷnandi Śivācārya (13 th c.) wrote the passage you quote, Indian authors of many different schools already spoke of living creatures that suddenly “burst forth” from wood, earth or even solid stone. The two examples he gives—“worms in the heart of trees” and “toads in the heart of rocks” —were in fact stock illustrations that Indian philosophers, physicians and lexicographers had been using for at least fifteen hundred years.

1. The four ways of birth
   The oldest Indian natural‐philosophical classification, found already in early Dharma- and medical texts, says that living beings are produced in four different ways:
   • jarāyujāḥ – womb-born;
   • aṇḍajāḥ – egg-born;
   • svedajāḥ – “sweat-born”, i.e. produced from fermentation, putrefaction, etc.;
   • udbhijjāḥ – “burst-born”, produced by splitting open the matrix that surrounds them.

Manu­­smṛti 1.46–47 (c. 2 nd cent. BCE – 2 nd cent. CE) gives the scheme; so do Mahābhārata 12.218.12, Caraka­saṃhitā Sū.1.26, Suśruta­saṃhitā Śār.1.10 and many others.

2. Frogs that burst out of rock
   As soon as commentators explain the word udbhijja the first example they give is “the frog that lives inside a stone (śilā­mandūka)”.

• Nyāya­bhāṣya of Vātsyāyana on Nyāya-sūtra 3.1.20 (4-5 th cent. CE):

 jarāyujāḥ manuṣyādayaḥ … udbhijjāḥ śilā-mandūkādayaḥ –  
 “Jarāyujas are men, etc. … Udbhijjas are the stone-frogs and the like.”  

• Śabara’s Mīmāṃsā-bhāṣya on Mīmāṃsā-sūtra 1.1.2 (c. 5 th cent. CE) uses exactly the same illustration.

• Medhātithi’s commentary on Manu 1.46 (9 th cent.) again cites “frogs which split rocks” (śilāṁ bhittvā mandūkāḥ).

• Varāhamihira’s Bṛhat-saṃhitā 54.19 (6 th cent.) lists as an omen of heavy rains:
yadi śilānāṃ bhidyamānānāṃ madhyāt mandūkāḥ kṛṣṭanti –
“if, when rocks are split, frogs come out croaking …”

The word śilā-mandūka “stone-frog” is entered in the oldest Sanskrit thesaurus, the Amarakośa (4-5 th cent. CE, 2.5.90), and in all later lexica; its very existence shows how familiar the idea was.

3. Worms in the heart of trees
   Wood-boring creatures are the standard illustration of the same class.

• Suśruta­saṃhitā Śār.1.10 (perhaps 1 st cent. BCE) while defining udbhijja explicitly says:
skandheṣu daru-romāṇi krimayaḥ jāyante –
“worms are generated in the trunks of trees.”

• The agricultural treatise Vṛkṣāyurveda of Paraśara (at least 6 th cent.) devotes an entire chapter to “tree-boring worms that originate inside the pith (kāṣṭha-krimi).”

• Jain scripture likewise: Tattvārtha-sūtra 2.34 with Pūjyapāda’s Bhāṣya (5 th cent.) gives “worms produced in wood” and “frogs produced in rocks” as typical udbhijjas.

4. Philosophical use of the examples
   Because “stone-frogs” and “wood-worms” were proverbial, philosophers routinely used them when discussing causality or spontaneous generation. Kumārila Bhaṭṭa (7 th cent., Ślokavārttika, Pratyakṣa 107-9) and Prabhācandra (11 th cent., NYāyakumudacandra 392) both ask how such creatures can be accounted for if the only cause of birth is sexual intercourse.

5. How old are the beliefs?
   The securely dated texts above take the motif back to the early centuries CE, and the classification itself almost certainly predates them. There is therefore no doubt that Aruḷnandi (13 th c.) was repeating a piece of “common knowledge” that Indian authors had been citing for well over a millennium.

In short: yes, India does have its own tradition of “entombed animals”. Sanskrit already possessed a special word, śilā-mandūka “stone-frog”, and references to frogs in rocks or worms in the heartwood of trees occur in medical works, law books, astrological manuals, Jain and Hindu philosophical commentaries, and lexica from at least the 2 nd-3 rd centuries CE onward—many centuries earlier than the passage you quoted.