trying again still get server error ... try this one little section at a time ...
Lets see if we got this right this time: vintage guy with a few old SeaDoos ... ya, that's it.
You have a good start: basic cleanup to see what you have and checks that will not cost a bunch of cash.
Battery is under the gray box in front of the engine, electronics stuff in the top of the box over the battery. Check for corrosion in the electronics.
I would guess it was left with fuel and oil in it. Drain it all. I recommend pulling the oil/fuel tanks out for cleaning; one of the ones I started with had some wicked sticky crap in the bottom of the fuel tank along with the check valve/filters that were supposed to be at the ends of the in-tank fuel pickup lines which had totally rotted off. Check operation of the in-tank filter/check-valve, I have seen them frozen open and closed ... should be hear the little ball rattle when shaken.
Go through the entire fuel system. Unless the fuel lines have been replaced, a 25 foot spool of automotive fuel line is pretty cheap. All of the older skis I have dug into, although looking good, the fuel line was vary hard and loose/cracked at the various connections. Fuel selector is a known problem, clean or better yet replace it. Clean out the fuel filter/water separator, insure the o-ring is in place ... check for leakage later. Recommended to insert an additional fuel filter just before carb, less than $5 for standard G2 auto filter is cheap insurance.
Clean and check carb; a can of carb cleaner "MAY" do it initially ... big subject, lotta threads regarding carbs here.