@cs22653291 MA plans are Medicare Part C . . . but that terminology essentially went away in 2006 with the creation of Medicare Advantage plans.
MA plans DO cover everything traditional Medicare (A & B) cover . . . but not in the same way. Your OOP cost under an MA plan may be higher or lower than traditional Medicare . . . especially if coupled with a Medigap plan.
Your pharmacy is normally not set up to file claims direct with Medicare (Part B) or Advantage plans. If they can't file claims they can't be reimbursed.
SOME pharmacy's can file Part B claims but that is normally used only for insulin.
If they are asking for your Medicare card it is possible they are confused as well.
This PDF explains a bit more about how Medicare handles vaccines.
Medicare Part B pays for one flu shot in each flu season, which typically runs from November through April, regardless of whether you’re enrolled in the original Medicare program or in a Medicare Advantage plan.
If you’re in original Medicare, the shot is free (no deductible, no copay) if you have it administered by a doctor or pharmacist who accepts Medicare assignment — that is, has agreed to accept the Medicare-approved payment as full reimbursement for providing this service.
If you’re in a Medicare Advantage plan (such as an HMO or PPO), the shot is still free. But you may be required to have it administered by a doctor who is in the plan’s provider network.
So your problem can be solved by going to one of your in-network medical providers - preferably, your primary care physician.
Pharmacist aren't normally in a medical provider network which MA plans have.