Cross-voting is not allowed. For a voter to vote in the runoff, the voter is required to vote in the Republican primary.
According to the Alabama SOS it’s only required they didn’t vote in the Dem primary:
I voted in my political party’s Primary Election. My party is not having a runoff. Can I vote in the other party’s runoff?
No. In 2017, the Alabama Legislature passed Act 2017-340. The act prohibits a person from voting in one political party’s primary election and voting in a different political party’s primary runoff election. Therefore, if you vote in a primary election and want to vote in a primary runoff election, you must vote in the primary runoff election of the same political party that you chose in the primary election. However, if you did not vote in a political party’s primary election and would like to vote in the primary runoff election, you may choose which political party’s primary runoff election you would like to vote in.
Ah, I got the law wrong. But it still prevents those who voted in the democrat primary from voting in the runoff.