When can a person join a Medicare Medical Savings Account (MSA) plan?

Since a Medicare Medical Savings Account (MSA) is a form of Medicare Advantage, you can join an MSA just like you would a Medicare Advantage plan: during your Initial Enrollment Period (IEP) when you first are eligible for Medicare and the Annual Enrollment Period (AEP or Medicare annual Open Enrollment Period).

As noted by the Centers for Medicare and Medicaid Services (CMS) in the document "Your Guide to Medicare Medical Savings Account (MSA) Plans" (when this document was still online):

You can join a Medicare MSA Plan during the following times:

When you first become eligible for Medicare during the 7-month period that begins 3 months before you turn 65, includes the month you turn 65, and ends 3 months after the month you turn 65. If you have Medicare because you’re disabled, you can join 3 months before and after your 25th month of getting cash disability benefits. The plan will tell you when your coverage will begin based on when during this period it got your request to enroll.
During the annual fall Open Enrollment Period (or AEP) that starts October 15 and continues through December 7th each year. Your enrollment will be effective on January 1 of the following year.

Note: The yearly deposit and the yearly deductible are pro-rated based on when your enrollment begins. Your enrollment will be effective no earlier than the first day of the month following your request to enroll.

(previous reference: https://www.medicare.gov/ Pubs/pdf/11206.pdf)
(older 2008 online reference: https://www.cms.gov/Medicare/ Health-Plans/MSA/ downloads/ MSAFactSheet-3-13-08.pdf )