If you have a rating of 40% disability for the Army and 100% VA, how does that usually work with pay? Will I get two separate pay checks?

There is some confusing answers. You get the 100% VA tax free. It doesn’t look like you are regular retirement eligible so you won’t get to double dip. Expect to get your Tax free VA and that’s it. So you get about 3k a month tax free.

This is how it's going to work.
You can only get 100% unless you are eligible for CRSC. You state that the Army says 40% so you're below the 50% threshold. So, this is how things are going to work. 40% is going to be paid by the Army, the other 60% comes from the VA. This is all due to concurrent receipt laws.
The good news, you're going to get 100%, the bad news, 40% of that is taxable. The worse news, that 40% is also eligible income in the event of legal action, etc. The bright side is, the 60% from the VA is tax free and exempt from legal action. Even better, since you're 40/60 you are eligible for Chapter 31 benefits.
But, check the laws they are always changing and you may have other circumstance's that you didn't mention. But I would only count on the 100% total.
I'm 40/60 also, so I do speak from some experience.

Hey Mike, the Rally Point community is awesome at alerting one to the fact that a question is extremely important and complex! You may want to think about communicating with a finance representative from your military branch of service. If not available, call the Defense Finance and Accounting Service (DFAS). Please also consider, if you are weighing a decision over accepting the responsibility of a military retiree ID card; there is more than simply taxes to think about. For example, what about your dependents, today or in the future? Have you weighed the value of Chapter35 (Dependent Education Assistance), and dependent eligibility for medical/dental insurance? Do you have or will you have children under the age of 18? Did you know that the VA may provide financial assistance for dependent children in pursuit of a higher education? Is the condition of your military discharge a mitigating administrative factor? Yes daunting, if I may suggest; for an immediate response to your military separation challenge, contact a local Veteran Service Officer (VSO), they may help you best understand your transition. Ultimately, plan your work, then work your plan! Think strategically, big picture, and lifelong! As a disabled Veteran/military retiree your administrative burden demands your best attention-to-detail! Take it seriously and get the right help.