I didn't read all the replies but I did do a google search for Kauai hardware stores and Kauai building supply and a bunch of ace hardware and other building supply stores came up. So you may have more options then just home depot. Most building supply stores carry tools also, although usually a little higher prices. There is also this place I found googling,
https://toolmasterhawaii.com/. Hopefully these links will work, you will have to change your location to Kauai home depot. Links will pop up at wherever you normally have set for your home depot location.
What type of cabinets are you building? Face frame or frameless? What are you doing for doors and drawers? This will change what tools you will need/want. I do face frame when I do cabinets, not a fan of frameless personally.
A track saw will do wonders for kitchen cabinets, that would be number one on my list no matter what. I just did a little searching at the home depot in Lihue Kauai and they don't have any track saws in stock. Then I started clicking on the ones that say they will ship to that store and it turns out your options are next to none. There was one track saw,
https://www.homedepot.com/p/WEN-10-...idewinder-Circular-Track-Saw-CT1065/310735605, that they would ship to the store there, it's compatible with makita tracks. But all the makita compatible tracks said no shipping to hawaii. But there was one track that they would ship to store or home, but that was a dewalt compatible track,
https://www.homedepot.com/p/POWERTE...atible-with-DeWalt-Track-Saws-71505/313524269. So unless there is more options I didn't find, you can't buy a track saw setup over there. If this is the case I would look into shipping your own, either ship the saw yourself and order the tracks from home depot. Or ship the tracks yourself and buy the saw there.
How does amazon work for shipping to Hawaii? Might be an option to buy from amazon and have it shipped to hawaii if there shipping to there is cost effective.
For face frame cabinets I like to cut the rails and stiles with a miter saw, they have a handful in stock there and can pick up a ryobi for 140 bucks,
https://www.homedepot.com/b/Tools-P...,1703?storeSelection=1705,1706,1702,1701,1703. I would probably upgrade the blade. I would rather have a decent saw with a great blade then a great saw with a decent blade.
Then I usually use a kreg jig for gluing everything up. They have those in stock there,
https://www.homedepot.com/b/Tools-P...,1703?storeSelection=1705,1706,1702,1701,1703.
If you want to get fancy you can do rabbits and dados for the cabinet joinery. I would usually do this on my table saw but that would add another expensive tool and dado blade set to your budget. You could do that or you could get a router and a few bits(check that they have a good selection of bits or have them shipped there) to do the dados and rabbits. You will also need a guide for that.
That should cover the basics of the carcasses of the cabinets. Now for the doors and drawers, this will depend on what your plans are. For the drawers I usually do rabbits, I don't have a dovetail jig and just never felt I needed to step up to that level(maybe someday). I wouldn't use a router freehand to do rabbit joints on short pieces like that. Might be doable but would get some snipe or chipping, I would want a router table. They have a cheaper one but not sure what all you would use it for,
https://www.homedepot.com/p/RYOBI-Universal-Router-Table-A25RT03/300234859. And at that point a table saw might be a more versatile tool to get. I also do a dado on the drawer to accept the bottom, this can be done with router or table saw.
The doors will be a little more tricky, depending on your design. You can do a simple sheet of plywood with a routed edge profile. Or do a raised panel but that will add more specialty tools. Then you have hinges, simple exposed hinges or concealed hinges.
What are you doing for counter tops?
What kind of timeline are you going to have? As in if you need a special screw or fastener or tool or bit and home depot or the hardware store doesn't have it are you there long enough to wait for one to be delivered?
Sounds like it's going to be a fun time.