Trigger that ends Eileen’s questline

Firing your gun while using a two-handed trick weapon isn't typically possible, but there is at least one weapon that allows you to do this:

The Rifle Spear has a built-in gun function while in its two-handed mode. I believe that you shoot it just like a regular gun, with L2. The item description includes the line "it is the only trick weapon with an attached gun," so it's possible that it's the only weapon in the game that can do such a thing.

There are also a number of trick weapons that still only require one hand even in their transformed mode, such as the Saw Cleaver, Saw Spear, and Threaded Cane.

In answer to your question: No, there is not an equivalent system.

However, you can use insight, the ingame currency, to buy items.

Sources:
"Items in Bloodborne are found in game [...] through trade with insight, or from chests. "
This site mentions nothing about traders, and a wiki would have mentioned so.