A chromatic instrument is like a piano. A diatonic instrument is like a piano that has lost its black keys. A chromatic hammered dulcimer is like a piano with some of the black keys missing. Which ones? You will have to ask the builder of the particular instrument.

The hammered dulcimer is by definition a diatonic instrument, and it'll always be that -- it's the nature of the beast. What we call a chromatic hammered dulcimer is essentially a diatonic instrument with several incidental notes (chromatics) added. 

Having all twelve notes means having a chromatic scale. It goes to reason, then, that you can play music in any key. Theoretically, if you have all twelve notes, you may, indeed, play a tune in any key. In real life, however, dulcimer players don't play notes -- they play patterns. Those patterns are the same for any of the native scales of the instrument -- A, D, G, C. You can learn to play a tune in D, but if it becomes necessary or desirable to play it in G, all you have to do is go up four rows of strings and hammer the same patterns. You may play the tune without ever knowing the names of the notes you're striking. But good luck trying to play a tune in, say, B flat.

Why not? All the notes are there, aren't they? 

Yes, there are, indeed, but they're scattered around the bridges, which forces you to apply different hammering patterns. Scattered notes also mean that there may be some big jumps between adjacent notes, which slow you down and make it next to impossible to play fast tunes. 

Another characteristic of the chromatic dulcimer is that because of space and design limitations, you may have every single one of the 12 notes on a given scale but not necessarily in every octave of that scale.  

If that sounds complicated, it's only because it is. I'll attempt an explanation here.

The hammered dulcimer lets you play in three or four major keys. These are typically A, D G and C. You can also play in the relative minors of these keys and various modes, but we'll forget about that for the time being to keep this discussion from getting messy.

To give you a wider range of playable tunes, makers have added extra notes -- incidentals. Those incidentals are usually found on the bass bridge, or on separate little bridges right and left. But since the basic layout of the dulcimer is the same, the player needs only to learn the position of those specific incidentals. There is no need to learn different patterns.

It's worth mentioning that there are other types of hammered dulcimers -- the santur, the cimbalom, the hackbrett, the yang qin, and so on, instruments that look just about the same but differ in the layout of notes. Also, some makers have attempted to redesign the dulcimer as a true chromatic instrument with all the notes laying in their natural progression next to each other, just like a piano. That gives the dulcimer an entirely different string layout, which requires learning new and different patterns of playing. You can't transfer what you know on a traditional HD to one of those strange birds. 

So, when a builder advertises his instrument as a fully chromatic, he means an instrument that contains all the incidental notes in every one of the three or four native keys of the instrument, but, as I said before, not necessarily in every octave. Usually, the incidentals are placed where they are most frequently needed -- in the middle range of the instrument. You will have to study the string layout of the particular instrument to know exactly what notes are there and where they are located.