I am trying to get the balances of a custom token on BSC (can be any BSC token – BUSD, WEJS, APX etc… you name it). To that end I have the following questions:

Is it possible to get the balances of a token without ABI?
If not, is there a way to automatically collect ABI information (like a Saas API)?

PS: I know BSCscan provides ABI for some verified tokens but it does not provide it for many of the traded tokens…
Is it possible to get the balances of a token without ABI?

No, it is not possible, but you don't need the full ABI of each and every token. The most tokens are made basing on some standards (presumably almost all tokens that you want to interact with are BEP20 [aka ERC20] or BEP721 [aka ERC721] compatible). Therefore instead of using their actual ABI - it is enough to use respective standard's ABI for basic and the most needed interaction such as getting balance or transferring tokens.
Therefore, just search the web for these ABI or simply compile from the libraries (for instance, OpenZeppelin) 'dumb' contracts of respective standards and use their ABI - it would be compatible with all tokens those following the standard (and all token makers, who need/hope for the third party support of their tokens do follow these standards).

If not, is there a way to automatically collect ABI information (like a Saas API)?

I'm not aware of this, but as I stated before - you can avoid relying on them. And token standards mechanism was designed with this very purpose.
Get balances of a custom token on Binance Smart Chain