Welcome, you will need this:
http://vmoa.net/VMX12-Service-Manual01.pdf save it on your computer and print out a copy for in the shop.
I suggest reading some of the threads about trying to return to the road a bike stored for a time. You may recognize some of the same issues.
Here is another thread:
New Vmax Owner FAQs....new members please read! RaWarrior has moved-on, but his thread still helps people get their bikes fixed. If you used the upper right-hand search function, on a subject concerning your problem(s), you're going to find info on it/them. Common issues are electrical, carburetion, second gear, and handling. With patience, proper tools, and by using the shop manual you now have, you should be able to solve anything. If you need to split the cases, but you choose not to do it yourself, member Damon in the Detroit area is about 200 miles away; member CaptainKyle is in Maryville TN something twice that distance, and Sean Morley in Derby KS is as-close as your local FedEx drop-off point.
Most members encounter carburetion issues at some point. Especially on bikes not ridden regularly, pilot jets become clogged. You can try chemical sprays and the 'peashooter' and the 'shotgun,' but if I suspect plugged jets, I drain the carbs to see what kinda crud comes out of each float bowl individually, and then it's 'tear 'em-off, & tear 'em-down.'
The 1st pic shows the starting enrichener pistons out of synchronization, the 'choke' function won't work properly. In this case, one or more of the pistons is probably frozen in the bore. Some of the parts for these carbs are no-longer able to be ordered from your friendly Yamaha dealer, so tread-lightly. This is one of the rare cases where you have to separate all four carbs, for access to the starter enrichener casting, which screws to the side of the carb. This is one of the pieces which is
not available new from Yamaha! A good soak in your favorite penetrating oil is one way to approach freeing up stuck pistons. A 50/50 mix of ATF and acetone is a good penetrant, but as acetone readily evaporates, and is flammable, use a closed container and do
NOT do it anywhere there is a pilot light, like a stove or a water heater. Flammable gases can travel dozens of feet to find a source of ignition!
The K&L kit here will save you significant $ over buying the OEM parts for the carb. I've used 'em for many years.
Two pairs is as-far as you often need to disassemble the carbs to clean the jet block brass jets. Note that the synch screws and the idle screw remain undisturbed, a big benefit. That is not to say that you don't need to re-synchronize the carbs after breaking them down this-far, and re-installing them. It just saves you the time, and possible screw-ups from an error during reassembly. Take lots of pics, from multiple sides, to see how things are as you tear them down, for ease of reassembly.