So there is good news and bad news (like always)
Bad: Neither A500 nor A513 appear in the prequalified base metal area, Table 1.2
If you get the 513 accepted, (the 500 is already specified, hopefully we can consider it accepted) you can qualify base metals as part of a welding qualification. You will want to specify grades to match the 500 in your request to the EOR
I would speculate that you will need a flare bevel joint with groove weld, a tee joint with a fillet weld, and possibly a square groove joint with a butt weld. Sadly, it looks like you may have to qualify 500 to 500, 513 to 513, and 500 to 513 for each of these (if needed)
Good news
The tests are very simple, manual destructive testing (no macro even), on coupons that are a few inches long.
sq. grooves are in Table 4.1, assy 4.1
fillet in a tee is assy 4.2C
In a bizarre departure from sanity, the flare bevel with groove is not drawn, but described in 4.6.3.2(1)(a)
Final bad news, getting coupons of the correct size may require some work, as you need a piece 2.5 in wide, 3" long, at least for the flares, cant see a minimum size for the others. I think we talked before about getting a larger tube and slicing flat coupons out of it, as I don't think you can get A500 or A513 in a flat section.
Watch the grades, as there are many as Scott mentioned, but you should be able to specify 1030 or maybe 1035 to match A500 Grade B
http://www.industrialtube.com/client/images/A513.pdfhttp://www.industrialtube.com/client/images/A500.pdf