Short answer: probably yes, but we don't know a lot about it.
We don't have enough documentation about the earliest stages of Japanese to be sure, but the consensus is that a bunch of the oldest words must have come from Chinese and other languages. It would be hard not to, since they were in contact all the time, and the original Japanese speakers came from the continent anyway. Linguist Bjarke Frellesvig, in A History of the Japanese Language, says that
It is beyond doubt that Old Japanese includes old loanwords from the languages around Japan—especially words relating to agriculture, seafaring, warfare, spiritual and religious life, government, and administration—but that we will not be able to identify many of them as loanwords on other than extra-
linguistic grounds. It is for example a strong hypothesis that Japanese iraka 'roof, roof tile' is a loanword, but we do not know from where. In other cases we believe that a word must be borrowed and can come up with several likely sources but cannot choose between them. It is, for example, very likely that the Japanese word for 'horse', uma, is borrowed and there are indeed words in surrounding languages which mean 'horse' and which are similar to uma, e.g. Early Middle Chinese *maɨ', Middle Korean mol, Mongolian morin.
These older loanwords would already be felt as "Japanese" by the time the country adopted kanji, and therefore they were classified as "kun-yomi". Other examples include:
- 栲 tae "cloth made from mulberry bark", which used to be tape, related to Austronesian tapa (same meaning).
- 釜 kama "pot", Old Chinese 坩 *khaam.
- 鎌 kama "sickle", Old Chinese *gryam.
- 絹 kinu "silk", Old Chinese *kwyans.
- 国 kuni "country", Old Chinese 郡 *guns.
- 梅 ume "plum", Old Chinese *hmay.
- 銭 zeni "money", Old Chinese *dzian.
"Bamboo" in ancient Chinese is thought to have been something like *truk > ṭuk, so I guess it does resemble take. But we can't be totally sure; some of these may be coincidences, some perhaps even borrowings from Japanese (or Korean or their ancestor, and so on).