Constitution of the United States, Article V:
The Congress, whenever two thirds of both Houses shall deem it necessary, shall propose Amendments to this Constitution, or, on the Application of the Legislatures of two thirds of the several States, shall call a Convention for proposing Amendments
How would the members of such a convention be chosen?
The method of proposing an amendment by way of Convention has never been used. Thus, procedural aspects such as how members of such a convention would be chosen have also not been settled:
In the modern era, scholars have debated various issues, including: (1) how delegates to the convention should be chosen; (2) whether Congress, state legislatures, or the delegates should set rules of procedure for the convention; (3) the vote threshold would be required to propose an amendment in convention; and (4) how voting rights on a proposed amendment should be apportioned among the states.
(Constitution Annotated, "ArtV.3.3 Proposals of Amendments by Convention")
As noted, we don't know. What do we know?
Basically, Congress would have to specify the procedure for selecting delegates, or at least a procedure for establishing how to select delegates, in the act calling the convention. (This is presumably why the constitution requires congress to call the convention when two thirds of the states ask for one rather than simply saying that a convention will happen when two thirds of the states want one.)
If the applications of the two thirds of states included identical proposed procedural rules for the convention then Congress would probably use those rules, but it might not. For example, if three fourths of the states were firmly controlled by one party, giving that party the power to ratify amendments unilaterally, while congress was controlled by another party, and the proposed procedural rules would allow the first party to control the convention, then congress might decide to alter the rules).
So, to answer your question:
How would the members of such a convention be chosen?
However congress specified when calling the convention.
