The 22nd amendment says you can, you just can't run again as president.
Wrong. 22nd doesn't say anything about the VP, and 12th amendment says you have to be eligible.
I assume you're referring to the final line in the 12th Amendment, which states,
"But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States."
You're right to point to this language, since it's one of the most significant arguments supporting the idea that a 2-term president would not be permitted to later serve as Vice President. However, there is some ambiguity in what the phrase "Constitutionally ineligible to the office of President" actually means in the context of the 12th Amendment. The most plain reading (in my opinion) is the one you posit, but there is an alternative interpretation - namely, that the constitutional eligibility referred to in the 12th Amendment specifically includes only those provisions found in Article Two, Section One:
"No person except a natural born citizen, or a citizen of the United States, at the time of the adoption of this Constitution, shall be eligible to the office of President; neither shall any person be eligible to that office who shall not have attained to the age of thirty five years, and been fourteen Years a resident within the United States."
In other words, it's plausible to read the 12th Amendment's restriction to mean simply that the VP must be a natural-born citizen, have attained the age of 35, and been 14 years resident in the US. Supporting this reading is the fact that the 22nd Amendment wasn't ratified until 1951, 147 years after the 12th Amendment - the 12th Amendment's framers couldn't have intended its language to preclude someone as ineligible on the basis of a future law they had no idea would be passed.
In the end, this is a question without a proper answer, since the issue has never actually come up. If a former 2-term president ever did run for VP and was elected, one could imagine his/her eligibility being challenged and likely litigated all the way to the Supreme Court, who would have to decide on which reading is the proper one.