Signed non-compete agreement with dog walking company, company was bought out. Does the non-compete still apply?

The non-compete may be deemed unreasonable and unenforceable in the first place, but to the extent that it was valid and enforceable before the buyout, it is probably still valid and enforceable.  If the "buy out" was a purchase of the original business's stock, the identical business - the party with which you contracted - is still operating, it just has different owners which changes nothing.  If the "buyout" was a purchase of the original business's assets, the non-compete agreement was almost certainly one of the assets which the new company bought and, barring a provision to the contrary in your employment agreement, is probably freely assignable.