In short, the answer is 'it depends'. A member of the Texas legislature cannot vote unless they appear for a vote voluntarily, so it all depends on the release conditions for the arrest, and whether or not the member actually wants to vote.
To be clear, Greg Abbott didn't say that he will have the politicians arrested, but that they will be arrested when they return to Texas. The quote can be found here (and elsewhere):
I can and I will continue to call a special session after special session after special session all the way up until election next year. And so if these people want to be hanging out wherever they’re hanging out on this taxpayer-paid junket, they’re going to have to be prepared to do it for well over a year... As soon as they come back in the state of Texas, they will be arrested, they will be cabined inside the Texas Capitol until they get their job done.
This is almost certainly a reference to the Texas legislature's ability to compel absent members to attend a session, rather than a threat to have them arrested on some criminal charge.
According to the Constitution of Texas, the Senate and House of the Texas legislature make their own rules on how to compel members to attend. Currently, both chambers use the same method: if there is an unexcused absence, the present members can send someone to arrest the absent member, and the present members decide on the conditions for release. The current rules also say only members who voluntarily appear are marked as 'present', it seems like it depends on how restrictive the conditions are: if they still have the option to appear for a vote then they can vote. If they can't or won't make it to a session under the release conditions then they can't vote, though this would defeat the purpose of compelling them in order to have a quorum.
The full text of Texas' constitution can be found here, and the relevant part is Article 3, which establishes the Texas legislature, in Section 10:
Quorum; Adjournments from Day to Day; Compelling Attendance
Two-thirds of each House shall constitute a quorum to do business, but a smaller number may adjourn from day to day, and compel the attendance of absent members, in such manner and under such penalties as each House may provide.
The most recent House rules of procedure, found here, explain how that could be done in Rule 5, Section 8:
Sec. 8. Securing a Quorum. When a call of the house is moved for
one of the above purposes and seconded by 15 members (of whom the
speaker may be one) and ordered by a majority vote, the main entrance to
the hall and all other doors leading out of the hall shall be locked and no
member permitted to leave the house without the written permission of the
speaker. The names of members present shall be recorded. All absentees
for whom no sufficient excuse is made may, by order of a majority of those
present, be sent for and arrested, wherever they may be found, by the
sergeant-at-arms or an officer appointed by the sergeant-at-arms for that
purpose, and their attendance shall be secured and retained. The house
shall determine on what conditions they shall be discharged. Members
who voluntarily appear shall, unless the house otherwise directs, be
immediately admitted to the hall of the house and shall report their names
to the clerk to be entered in the journal as present.
The rules for the Senate, found here, are similar, as seen in Article V, rule 5.02:
Rule 5.02. Two-thirds of all the Senators elected shall constitute a
quorum, but a smaller number may adjourn or recess from day to day and compel the attendance of absent members (Constitution, Article III, Section 10). In case a less number shall convene, the members present may send the
Sergeant-at-Arms or any other person or persons for any or all absent members.
The Senate shall determine upon what conditions they shall be discharged. Members who voluntarily appear shall, unless the Senate otherwise directs, be immediately admitted to the floor of the Senate, and they shall report their names to the Secretary to be entered upon the journal as present.