It can be done, but it might not be easy.

Odds are high in an older home that the chimney above that fireplace is brick. Removing enough for your cabinet while leaving enough [actually, building some sort of structure] to support the brick above is probably a job for an expert, or at least a very skilled DIYer...

I agree that it probably will not be easy. My father is an experienced DIYer and has actually removed a majority of 2 different chimneys in his mill house. He accessed them through the attic and got as many bricks out as possible. We were thinking of doing the same with ours and then continuing the brick removal when we cut the hole in the wall for the cabinet. That way there would be no bricks above to worry about and we can close the open space from the cabinet and the attic.
Would there be any structural issues to the house due to the space left by the removed chimney portion?
We plan to use the bricks in the yard for raised flower beds.

I'm not a mason or engineer, but what I'd want to see is a structure of some sort to carry the weight of any bricks left above the opening down to a weight bearing surface below.

Something as simple as 2x4 framing at the sides of the cabinet opening would probably be fine, and it sounds like you've already given this a lot of thought and have more hands on experience than I do with these issues.

Since I live in earthquake country [everyone does, acutally: its just a matter of frequency], I do worry about a chimney taking a tumble during a during a tremble.

Love the idea of re-using the bricks. I want to incorporate some of the materials from my great grandfather's homestead cabin in my new home. Materials that carry the weight of memory are much more interesting...

Make sure the chimneys(s) are not connected to some more recent appliance that needs venting: hot water heater, boiler, furnace, etc. Check in the attic, sometimes a homeowner or installer may access the chimney there as being less trouble than cutting a new hole in the roof.