In this sentence, "It is critical that the supplier keep customers apprised of...," should the word "keep" be singular or plural?
It is written in the Subjunctive Mood, used to emphasize, in a formal way, the importance of a direction or order.
"The doctors ordered that his father remain in bed for a week, and that he return to work only when he is fully fit."
Your sentence is a directive, termed 'critical', and so its importance is emphasised by the use of the subjective, 'keep' instead of 'keeps'
It is critical that the supplier keep customers apprised of...
It has nothing to do with singular or plural. As written, this is a mandative subjunctive construction as evidenced by the use of the plain verb-form "keep".
There's no reason why you can't drop the subjunctive and use the non-mandative declarative with the tensed form "keeps" -- the only difference is one of formality, with the subjunctive being the more formal option.
Incidentally, 'subjunctive' is NOT a mood form but a construction type recognisable by the use of the plain form of the verb.
And it's NOT a directive either, though it does invoke the concept of 'compliance'.
Would Mr.Matthews care to explain why "The doctors ordered..." is not a directive, given by the medicos?