She does use the ammo so it does need to be re-supplied, but the gun itself won't lose condition or break (unless you accidentally shoot it, which can happen).
Veronica however prefers punching, and as such, she might be better at it. Can't say for certain, I just remember telling her to use her pistol one time and she immediately started complaining about how much she hated it. Lily and Veronica will both prefer melee combat, and to be honest I see no reason to discourage Lily from doing so since she runs fast as hell, has lots of HP and can close the gap on the enemy quick.
For the most part though, only give companions new weapons if you have cash to burn. It can be kind of expensive and most of them have good weapons as their defaults anyways. Lily or Veronica are worth upgrading since upgrading melee fighters is as simple as giving them a new weapon (no ammo needed), but aside from that everyone has good default weapons. The exception is Cass; give her something cause he default weapon svcks balls.