This transmitter was designed around the Taylor T21...which is a 6L6 with a six-pin base. Another, similar bit of 6L6 cloning was the Raytheon RK49. (There's a top-view photo of the little transmitter in the "Radio" handbook for 1940 -- I'll see about scanning it this evening.)
There is a reason for this: the 6-pin base has wider spacing between pins and will run higher voltages with less chance of an arc-over. Especially important at radio frequencies. The Raytheon appears to have a ceramic base. I'm not sure what Taylor used.
In most HF applications that aren't pushing the tube too hard, you can substitute a 6L6G/GA/GB for these tubes without much trouble.
BTW, these things can become quite valuable -- there's a Utah "Junior" single-6L6 transmitter, albeit a very fancy one, about to change hands for over $500 on an auction site.
Update: Bruce, W1UJR, points out in comments that it's more fun to restore a Utah Junior than buy one bandbox-new. He's right -- and has done a first-class renewal on his.