Can a person that can pass a back ground check buy a gun in a state other than where they live?

check with both states laws,
are you talking long guns or pistols?
for long guns, generally if the gun is legal in your home state, they will sell it to you,
some shops are not familiar with all states regs, and try only to sell to bordering states that they are more familiar with,

AR15, it's legal in my state, but I am having trouble finding them in stock.
I have to go on a road trip for work and wanted to know if I could buy one if I find one on my travels.

#1...

Easiest way to do it...every state law is different, so you'll have to ask...and I have found some stores interpret state laws differently, I reside in Michigan and was working in Utah...Cabela's in Utah would sell me a rifle over the counter, Sportsman's Warehouse wouldn't...

You are only able to face to face purchase firearms in your own state and those contiguous to it. Any state not touching your own you'd have to have it shipped to an ffl in your state.

thanks for the info from everyone, much appreciated

Go to your "local" FFL dealer - they will walk you through it.

For a fee, of course !

Or, contact one of the big, online dealers (like Bud's) and I'm sure they'll help you do what you need to so they can make a sale. Some have a list of FFL's near you.

Unnecessary... Any FFL knows the drill, just order/purchase the gun anywhere, have your local FFL of choice mail/fax the selling party their information and FFL copy, then have it shipped to your local FFL to pick up at your convenience.

Right on.. Most sellers will ship the gun with a copy of their local FFL dealer they went through to the FFL dealer you wish to have receive it for you after they receive a copy of your dealer's FFL and shipping address. Be aware.. a lot of FFL dealers working on YOUR end of the deal will NOT accept a gun, and particularly a handgun unless it is accompanied by a current, signed FFL along with the shipment.