I am analysing piping connected to a couple of tanks that are 13 metres in diameter with nozzles 2/3 of the way up the sides (therefore API 650 Appendix P is apparently not recommended - not a hugely useful appendix a lot of the time ). I do not have access to FEA software and am looking for guidance as to acceptable/reasonable tank nozzle loads I can use (realising the tank vendor will have the final say).
Would the use of the WRC 107 module with ASME VIII stress summation provide a reasonable guide as to local tank stresses at the nozzle?
Can I still include API 650 nozzle flexibilities in the Caesar input for a non API 650 tank?

You're right in stating that API-650 Appendix P isn't applicable to your nozzles. Without performing FE analysis, your best option is to try to use WRC-107, as you stated. Of course, this assumes you don't exceed any of the dimensional limits that would put you out of the scope of the 107 bulletin. If that happens, then you'll have to make some assumption, or bite the bullet and have someone do an FE analysis for you.

What ever you do, make sure you document it. That way, if someone disagrees with you, they can offer a better solution. At least you told them up front.

You're right here also. But if you read the references for this appendix, you'll find that Appendix P was developed from two technical papers, discussing the instrumentation, testing, and subsequent FE analysis of two full scale tanks. The limitations (in the appendix) exist because of the limited amount of testing.

You also questioned if the API-650 flexibilities are applicable to your situation. I'm going to lean towards "no", and suggest the WRC-297 nozzle flexibilities (again watch dimensional limits).

I hope this helps. Maybe someone else has something additional to offer?