I worked as a trade plater for 2½ years and the firms (or individuals) that I collected vehicles from never required me to prove that I held the relevant driving licence, nor that I was covered by 3rd party insurance. (Indeed, I was often expected to accept a vehicle which would have been illegal to drive. It wasn't unusual to be told that "You don't need to worry about the MOT, mate" or "OK, yes, it has got four bald tyres but you'll be alright". People got quite uppity when I refused to accept such vehicles!).
However, should a case come before a court, it would be up to that court to rule upon what constituted "to cause or permit" (unless there was a relevant precedent that could be referred to)
Section 87(2) of the Road Traffic Act 1988 states "It is an offence for a person to cause or permit another person to drive on a road a motor vehicle of any class otherwise than in accordance with a licence authorising that other person to drive a motor vehicle of that class".
Similarly, Section 143(1)(b) states "a person must not cause or permit any other person to use a motor vehicle on a road or other public place unless there is in force in relation to the use of the vehicle by that other person such a policy of insurance or such a security in respect of third party risks as complies with the requirements of this Part of this Act"
My view is that a court would rule that, should a driver accept a vehicle when he didn't hold the relevant licence and/or insurance, it was the firm that sent him to do the job, rather than the firm he'd collected the vehicle from, that had 'caused or permitted' the offence. i.e. it's up to a firm employing trade platers to check that a driver holds the correct licence and to ensure that either the driver or their own firm has taken out appropriate insurance cover.
However (unless there's a binding precedent) a court must consider each case individually, so there could possibly be certain circumstances where the firm handing over the vehicle could be held liable though (although I certainly can't think of any such example).