I have no information on the content of service records supplied by the MoD, other than what is on the veteran's agency web site, and that's bit ambiguous "request a full copy of record" then "MoD will provide an extract" suggest you ring the help line and ask for clarification. Before applying for records there's a number of things you could do.
1] Ask your parents, and other relatives if anyone has his pay book. All WWII British Army personnel were given it on discharge with their discharge certificate pasted in the back. The certificate gives his army number, rank on discharge, cause of discharge , campaigns & service abroad, sometimes medal entitlement [can be blank] and conduct. If not ask if anyone has his service number anywhere.
2] Write/contact the Royal Logistics Corps Museum [they don't hold personnel records] to see if they have any details on the movements of his company, see the RASC's page of the regiments.org site at http://www.regiments.org/regiments/uk/corps/RASC.htm
, for museum address [use 2nd link] or see http://www.waggoners.co.uk/
, if they can't help try the National Army Museum or the RASC Association.
3] See if your local library can get hold of any of the books listed under Bibliography on the regiments.org page above.
4] Write to the MoD Medal Office [will require letter of authority from [direct legal] next of kin] and request details of his medal entitlement [give as much personal information as you know, date & place of birth, service number if known, date of death [copy of certificate may be needed]. If his WWII medals were never claimed then next of kin are entitled to claim them. See http://www.veterans-uk.info/index.html
, LH menu, click on medals / badges / emblems. See if you can identify the persons who are legal next of kin before you, use "certificate of kinship" from vets agency site.
He should certainly be entitled to the 1939-1945 Star and the War Medal, then depending were else and dates he served one or more campaign stars and the perhaps the Defence Medal, see http://www.stephen-stratford.co.uk/wwii_medals.htm
, scroll down for medal descriptions and qualification.
5] Last resort apply for service record.