Probably "while", to emphasize that the holiday (US "vacation") lasted for a while. We could also use "when". With a verb of state, the word "as" would tend to imply a causal relationship, i.e., being on holiday caused Tom to come down with the flu. If we used an inceptive verb with a more perfective aspect, for example "Tom came down with the flu as we began our holiday", that would work quite well with "as" (meaning at the time), or we could use "when". "While" would not work well for such a verb / aspect.

To my knowledge,all the alternatives are fine because my Grammar book says" we can use as, when and while to mean "during the time that". What do you think of it?

I agree with Phil. WHILE or WHEN but maybe AS will work. What about "Tom came down with flu as we were on holiday," meaning that Tom is an annoying person who always gets ill at the start of something! "Tom didn't get out of bed as we were on holiday." Yeah?

While