Short answer. The short answer is yes: because all 7 scale degrees are represented by one of the chords, it is possible to make a harmonisation that only uses I, IV, and V. This is pretty close to how a typical treatise of music theory starts: start with these harmonies, and add more as need be.
Long answer. The longer answer is no. Although you can string together a progression of I's, IV's, and V's to accompany any melody, this does not necessarily mean that the accompaniment sounds good. This has to do with structural functions that each of the harmonies imply.
Concretely, a typical musical phrase will develop (on the large scale) as I IV V I or a variant thereof (e.g. I ii V I or simply I V I). There might be some other chords in between, but most likely all earlier incarnations of IV and V are in some inversion or otherwise have a weaker structural function. If you arrive on a V, in a lot of situations the only way out is to go to a I afterwards. Similarly, the strongest tendency for IV is to go to V (although back to I is also ok).
This is where other chords start coming in handy: if you move to IV or V too early, your melody is over and has to resolve. But sometimes you want to stretch it a bit longer, and then chords like vii˚, vi, iii, ii, and inversions of all triads become more important.
So what now? The harmonies I, IV, and V are certainly enough to get you started. But as you practise, you will start to notice that you want to do more, and that is probably the right time to add other harmonies to your repertoire.