Nearly no wallets ask for your ID or bank account. Only exchanges ask for your bank account, which are not wallets as such (Coinbase has a "wallet" but that's essentially an addition to their exchange).
As for wallets where you own the Bitcoin properly, which is what you'll probably want for your trades, you could try software wallets like
Electrum or if you don't mind taking up a lot of space and bandwidth you could try a full node like
Bitcoin Core.
You also might want a hardware wallet like a
TREZOR if you're not tech-savvy or you're prone to malware.