Harris will be able to serve two terms. The 22nd amendment says:

No person shall be elected to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of the President more than once

The second clause applies to vice presidents taking over, but since there’s less than two years left in Biden’s term, it won’t come into effect.

I’m not a legal expert so take this with a grain of salt. Assuming the Constitution is followed as it currently written…which is a big assumption…

The two-term limitation comes from the 22nd Amendment. This amendment is, in it’s entirety, quoted below:

Section 1

No person shall be elected to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of the President more than once. But this Article shall not apply to any person holding the office of President when this Article was proposed by the Congress, and shall not prevent any person who may be holding the office of President, or acting as President, during the term within which this Article becomes operative from holding the office of President or acting as President during the remainder of such term.

Section 2

This article shall be inoperative unless it shall have been ratified as an amendment to the Constitution by the legislatures of three-fourths of the several States within seven years from the date of its submission to the States by the Congress.

Concerning Section 2, it was ratified so it is law. Not important to answer our question.

If Joe Biden resigns or is permanently incapacitated right as I post this, then Kamala Harris would become acting President until his term ends on January 20th, 2025 for 157 days, 158 if you count today, which is a far cry from two years. (I believe that she was designated Acting President for a few hours while Biden was getting a colonoscopy, so add that to the tally maybe probably not.)

So it looks like, if I read the bolded section correctly, she would be eligible to be elected a second time if she took over from Biden should he be incapacitated or resign.

Edited based on feedback.