I recently went through this and did a little shopping before finding a credit union to try out. I had originally wanted to open a checking account with Knoxville TVA Employees Credit Union because they financed my car (so I already have a savings account) and they're building a branch here in Johnson City. Before deciding to wait on it to be built I called the main number in Knoxville to see what their policy was there and they told me no guns allowed. Dunno if it'll be the same here, but I didn't bother waiting to find out.
So I ended up choosing Northeast Community Credit Union specifically because they didn't have their locales posted and nobody said anything when I opened my account a couple weeks ago.
I also bank at Suntrust and Carter County bank and neither of them are posted. Out of the three the only one I haven't OCed to is Suntrust, but I've been biding my time before OCing into Suntrust. I'm this || close to closing my accounts with them after getting a $17 service fee for having the upper tier checking account that I had. I've since downgraded that account, but I'm waiting to see how I like NCCU before dropping them completely. When I make my decision with NCCU I'm going to OC into Suntrust and if anyone says anything immediately ask to close my accounts.
So I can't speak for any of the credit unions in the Loudon Co area, but I suggest shopping around-that's what I did. Just drive up to the building and see if it has a sign. If not, walk in OCing and see what happens. Otherwise, Suntrust isn't a bad bank to deal with as long as you don't mind keeping the lowest tier checking account to escape the fees. In fact, their online banking system is actually pretty fantastic.