Can my military experience (MDMP) get me Lean Six Sigma or Project management certification?

Yes you can. All operations you did when you were in consider project management. DO NOT think operations in the Army is the same as the civilians. Operation management in the private sector is a glorified maintenance/supply/"sir fix a lot" officer. If you need help of translating your experience in to PM and get qualified to the PMP course, message me and I will walk you through. I just finished my PMP.

CPT Coutu,
Military experience can absolutely be used for practical experience when applying for the PMP. When counting practical hours remember to count the time spent throughout the entire project or operation from initiation thru planning, execution, controlling, and closeout. Just thinking about MDMP doesn’t cover the range of project management, and will make it hard to get the 4,500 hours (with a 4 year degree) required for PMI. If you planned a deployment you would count time from receiving the WARNO, planning the deployment, actually deploying (execution & monitoring) and through RSO&I (closeout) which can build up hours quickly.
For the PMP you still need 36 classroom hours from an accredited provider. This is essential as it helps you translate military operations into civilian speak, and it teaches you how PMI views project management and breaks down project management with 5 processes and 10 knowledge areas.
eArmy learning has online classes that are certified by PMI for free. But I recommend going through Syracuse University's Veterans Career Transition Program as SSG Erickson mentioned. They provide the same classes, but when you complete the classes through them they will pay for one free PMP exam. While I’m not a fan of online courses these do a good job explaining how PMI sees project management and explains the Project Management Book of Knowledge (PMBOK) which is how PMI views project management and is the basis for the exam questions.
The hardest part of the PMP exam is to answer question according to how the PMBOK/PMI project management process, and not according to how you would or have done things.

I was a PMP and Army Certified Lean Six Sigma Black Belt; also Lean Six Sigma program manager for SDDC. Retired 2013.

Project Management Institute (pmi.org) developed national standard for project manager certifications. PMP requires project management education, project management experience, and passing PMP exam. Military experience may count for required project management experience and some training (on-line or classroom) may count toward the project management education requirement. Commercial training organizations offer week-long classes that guarantee passing the PMP exam. Last time I checked they cost around $5K. PMI chapters often hold PMP exam preparation seminars. The leaders are always local PMPs. They may cost from $500 to $1,000 (usually discounted to PMI Chapter members). Many employers will pay for test and some will pay for a prep class.

There's no national standard for Lean Six Sigma certifications that I know of. Colleges, Universities, non-profit organizations, the Services, and businesses offer Lean, Six Sigma, and Lean Six Sigma certifications. (I see you're in the Lean business, so I believe you may know more about education opportunities than I.) IMO, best practices require education and successful project completions. Army Black Belt (BB) course was 4 weeks. As an Army Guard member you might get your unit to allow you to attend the Army BB training. Army Certification required completion of BB training and 1 BB project. Some corporations require multiple projects or a certain amount of hard savings. Some schools grant certifications using only an academic project. Military process improvement projects are perfect for Lean Six Sigma process application.