Does the badge affect wireless charging?

Only the badge is concerned here, since it is the only object that goes on a phone. A card in a wallet and a keychain on a keyring change nothing about charging.

The badge is thin, but like anything on the back of a phone it can get in the way of wireless charging if it covers the charging area.

The fix is placement. Wireless chargers (and MagSafe) send power through a coil in the centre of the phone's back. Stick the badge in the lower third of the back, clear of that centre area, and wireless charging keeps working normally.

Sliding it sideways will not do: the charging ring is almost as wide as the phone itself, so there is no room left beside it. Downward is the way out.

If your phone ever warns that charging is blocked, or charges slowly, move the badge a little further down. With a wired charger, the badge makes no difference at all.