How would you realise an intonation system different from equal temperament on a modern trumpet? As a pianist, I can see how that'd be done on a stringed instrument or a trombone, but aren't the pitches on a trumpet fixed in the same way they are on an oboe/saxophone, with just a different valve system?
Embouchure and valve manipulation can influence the exact pitch of a tone on all wind instruments. For proof, just listen to any jazz trumpet player!
The effect is larger on some instruments than others, but it's easily large enough to switch between a just and a well-tempered third, for instance.
The pitch difference between a note name in Just Intonation and the same note name in 12-EDO is very small. A trumpet player wouldn't need to use valves to adjust from one intonation system to the other. It can all be achieved via the embouchure.
When you play Just-Intoned notes you listen hard and adjust until the harmonics of the note you're playing "lock in" with the notes other people are playing. This effect is much better if you're in the room with the other musicians. The difference between a 12-EDO perfect fifth and a Just perfect fifth is that the Just interval sounds locked, it doesn't swim slightly like the 12-EDO version.
Playing different intonations in Ensemble is relatively easy, but playing the exact right frequency without reference to other players is very much harder.
The basic answer
Each of the tubes that comes out of a valve on a trumpet can be moved in or out a little to handle certain changes in intonation. Just taking a cursory look at the numbers, so long as you manage to keep them in the right intonation in one register, they will be in tune in all registers.
That said, the trumpet's open notes (C4-G4-C5-E5-G5-C6) are already in just intonation. This is because they rely on the overtone series to work.
How do the valves work?
A trumpet has three valves: 1 - a whole step down, 2 - a half step down, and 3 - a step and a half down. Trumpets arrive at different pitches by going to a pitch on the overtone series (C-G-C-E-G-C above) and then going down based on the valve configuration. So, for example, D4 is G4 overtone + valves 1 and 3 (for a total of 2.5 steps down).
Just intonation works because E:G (valves 1 + 2) is the same interval as A:C (also 1 + 2). This is both true in music theory (both are a minor third) but also true mathematically. Meanwhile F (valve 1) is a different valve from B (valve 2), meaning that these two can be tuned accordingly without a problem. That said, you do have options to tweak the tuning while in a performance. If you need a different tuning for A (say, you wanted it to be a perfect fifth above D instead of a Major third above F), many trumpets offer a small ring that will let you adjust the tube length of valve 3 at will.
I think the answer has to be clearer.
First: many trumpets *) have adjustable sliders on first and third valve. These allow the player to adjust the length of the tubing, that is lower the tone slightly. The first valve slide is operated with left hand thumb, the third valve slide is operated with the left hand ring finger. If you look at a picture of a trumpet there typically is a "ring" for third valve and a fork or sometimes trigger for first valve.
Second, but actually most important: the intonation of tones can be influensed to some degree by the lips and embouchure of the player. Most tones can be played slightly higher or slightly lower depending on the "tension" in the lips and other factors such as tongue position or opening of throat. This has to be learned through years of practice - one of the reasons why beginners on the trumpet are so "not in tune".
Third: on some tones it is actually possible to play with different "fingerings", which can affect intonation. A high G can as example be played both with no valves down and with 1+3. A middle of clef A is usually played 1+2 but can be, more or less, played with a 3 (out of tune though).
So, as the player anyway adjusts the intonation of every note it allows, with practice, a just intonation.
*) the description is for a more or less typical piston valve trumpet, search Youtube for "trumpet sliders". Rotary type trumpets come from a slightly different tradition and work slightly different. Some in addition have extra keys in the tubing which can be opened to modify intonation or to easier play some notes: search Youtube for "rotary trumpet".