What kind of weapons can pets use?

A pet can use any weapon provided it is not NO TRADE. A pet must also be of at least the required level of any combat level effect ("proc") or the weapon won't proc (and you will see a message telling you that).

The delay of a weapon is always ignored, as the pet will always fight with their own natural delay regardless of weapon. Only the damage of the weapon is considered, and will only increase the pets damage when the weapon damge is more than half the pets own base damage without weapons.

For very low level pets giving it a high damage two-hand weapon such as a two-hand sword, scythe or halberd (even a rusty one) can increase its damage slightly. After the first couple of tiers of pets most weapons don't have enough damage to increase the pet's damage over its natural damage. The main advantage of weapons to higher level pets is the "proc" that a weapon does.