Paul Corkery from the organisation said a points system, intended to favour fans who have attended the most games, had left fans "confused" and "annoyed".
He said in some instances fans with fewer points had been given tickets when people with more points than them had missed out.
Successful applicants have been informed by email.
Mr Corkery told BBC Radio Wales' Good Evening Wales: "A majority of people who've been to most games seem to have got tickets okay."
But he added: "[After a] certain amount of points have been taken into consideration - it seems to be a bit of a lottery.
"It's a bit confusing then for people who are a bit annoyed about that."
Playing at smaller venues meant Wales had a lower ticket allocation than most of the 24 teams in Euro 2016.
The Football Association of Wales (FAW) received more than three times as many applications for Euro 2016 group matches as it had tickets.
The FAW said: "While we are pleased to see many celebrating that they shall be watching our country in Bordeaux, Lens and Toulouse this summer, we are also sympathetic towards those fans who have been unsuccessful.
"When the ticket application window closed, UEFA shared the list of members who had applied for tickets with the FAW.
"The FAW implemented its loyalty system and prioritised the lead applicants before sending the data back to UEFA."
Ticket allocation for the Euro 2016 finals was a "lottery", the Football Supporters' Federation has said.