Although every LGS has different rules, I don't know of any that will hold a gun indefinitely for you.
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One LGS told me they've held guns for deployed military for years while the owner was deployed.
My guess would be the length of time they happily hold a firearm you paid for would be case by case, factoring in the length of time and situation.
Once you pay for a gun, it is your property. If you take an extra long time to get the permit and pick it up, the shop really doesn't have much to complain about other than having to make space for it and count it in their inventory occasionally.
The shop can't legally make you pick up the gun if you haven't gone through the acquisition permit process. In fact, it's their job to ensure you do that BEFORE they deliver the firearm to you.
As long as you maintain communication with them so they know it's not abandoned, most of them will work with you however long it takes.