Is it possible for a timpani player to play multiple notes at the same time such as a guitar or piano player would do?
One timpano will play one note a one time. The sound has several harmonics which can be heard, but that's the case with many other instruments too, including guitar.
Timps have pedals which re-tune the basic note,by re-tensioning the heads, but those notes get played separately. To play 'a chord', there needs to be two, probably three timps, all tuned appropriately, played simultaneously. And that's not easy for one player, although not impossible.
I think it depends how just how similar to guitar/piano you are expecting.
I think this video gives a good idea of what a player can do...
With two sticks and four drums available the player can obviously play at least two different tones simultaneously using two drums. Depending on the desired rhythm a drum roll might be used which would be like a tremolo on piano.
If three tones were wanted, I suppose the reality would be a kind of broken chord pattern. The volume decays quickly so it probably will be hard to hear those as a proper, simultaneous chord.
But I think the really important concern is about the harmonic clarity. You could get a two-tone tremolo easily, but the result may be pretty noisy, not as clear as what the piano would produce.
BTW, I tried to find videos searching 'timpani chord' but didn't find anything. But I imagine if you look up more timpani performance videos you will find something close to what you are interested in. Maybe try searching 'timpani solo'. Here is another example that may help answer your question...
One kettle can play one note. Give a good player four kettles, it wouldn't surprise me in the slightest if (s)he found a way to play them all at once! Musicians are endlessly ingenious.
I wouldn't recommend WRITING more than two-note chords unless it's a virtuoso piece and you've consulted the player though.
(I don't THINK there's any way to coax two pitches out of one kettle, in the way that windplayers do 'multiphonics'. But, again, I wouldn't be all that surprised to be proved wrong.)