Having joined the Night's Watch, he forsook his rights to any land or titles. So why is he on the council with all the lords?
Sam was actually appointed Grand Maester (senseless as that may be...), which grants him a place in the small council, which presumably would also grant him presence (and a vote?) in what I assume is a short-staffed Great Council.
In the small council, we see Sam all dressed in white, as is custom for Grand Maester, and Tyrion addresses him as "Grand Maester" too, just so we can make sure.
He was first in-line to succeed his father as head of House Tarly. Now the Night Watch's oath makes him abandon this claim, but it has also been made clear that under Daenerys' rule, he has been excused from his Night's Watch rule, because he was allowed to stay with Gilly, who is pregnant of his child (a big nono). It is made fairly clear everybody follows more or less Daenerys' decisions (Gendry also being there, Yara explicitly saying so). So it's fair to say according to this he also can re-claim his titles and thus would be Lord of Horn Hill. Obviously he later throws of this title again because he re-joins the citadel.
Note that Horn Hill is one of the mightiest houses of the Reach, maybe even the mightiest in-show since the Tyrells are gone and Florents seem to be irrelevant, so he could very well be seen as representative of the whole Reach, making him having one of the more important votes.