Firstly, welcome to rootschat SL.
There are a lot of people unhappy with this company, and claim that the information on this site which you need to pay for, is available elsewhere for free or as part of a (useful) subscription.
I am rightly or wrongly, assuming you are referring to WW2?
In order to obtain your father's service number there may be other avenues you can try - all of them are free. You could contact the medal office at Imjin Barracks in Gloucester to check on his medal entitlement. The correspondence you receive from them should give his service number - they may supply the service number to you anyway if you explain the reason you need it (this is just a guess on my part so may not be possible). This may be do-able via email.
You could apply for his service records but will need to provide identifying information such as date and place of birth. As your mother is still alive, she is entitled to his service records at no cost (it would cost you £30). You could do the form filling on her behalf and all she would need to do is sign the form.https://www.gov.uk/get-copy-military-service-records/apply-for-someone-elses-records
Let us know if you need further information or advice on applying for records. I would have thought that the British Legion might have access to service numbers, but once again that is not based on any knowledge on my part (it just seems logical).
There are plenty of military experts on rootschat who should be along soon to offer you further help, and correct any errors I may have made.