A108 isn't a list prequalified material, so as you noted, the procedure must be qualified by testing.
You can always qualify the procedure in accordance with Clause 4 provided you can get the material in the appropriate form. In your case, A108 is bar stock, so getting it in the form of plate is out of the question.
If the parts you are welding are not required to comply with AWS D1.1 you can use D1.1 as the basis of qualification, but don't include a statement that it meets D1.1 in your certification statement. You may be able to find a plate material that has the same mechanical properties and similar chemistry that would be a satisfactory substitute for qualification purposes.
That being said, I would suggest you use AWS B2.1 as the basis of qualification since it does include ASTM A108. It is listed as a M1 base metal. That allows you to use any M1 base metal (including ASTM A36) to qualify your procedure. You can expand the list of welding variables to include those included in D1.1 (arc voltage, amperage, etc.) if you wish to. Then, if you have a customer that wants the procedures to be qualified per D1.1, you can submit them as an alternate to D1.1 as allowed (with the Owner's approval) per clause 4.1.1.2 (D1.1-2006 as an example).
Best regards - Al