Members of the legislature can become legislative committee members.
The US House of Representatives and the US Senate each have their own rules for how committee members are assigned to committees. For instance, House Rules allow members of the house to serve on up to two committees and four subcommittees. In the Senate, the number of committees on which a senator can serve depends on the type of committee. (They have class A, B, and C committees.)
But as far as who can serve, any of the elected members of the legislature are eligible to serve on committees.