Mark, the TNA website is a tricky one. Not for the faint hearted.

I have managed to try a test search of a common name and, looking at the images, what is shown are photos of hundreds of typed sheets of paper indexed alphabetically by surname and forename. Information on each record is minimal and varies, but all have a name and service number, only some have dobs etc. and if there are several men with the same name there is no obvious way to see which is the serviceman you are looking for .... or, have I missed something?
If the OP is simply looking for service records, which it seems they are, then it is not necessary to supply the service number on the application. It is easier to request that the MOD search for the name and cross check with dob, pob and any other information that the OP can supply.
It might be useful if you could provide links to online service numbers for those who served in WW2 if you are aware of such lists, as this is a question which is often asked here on rootschat, and I'm sure it would be something of great interest which would be "stickied" for others to use.
My suggestion of contacting Imjin was just that - a suggestion, and another possible way to obtain a service number, whilst also checking medal entitlement (if any), and checking to see if, when and where any medals were sent. Some men did not claim their medals and descendants may wish to do so. It can all add to the big picture.