Goals from Nathan Smith and namesake Connor against Forest Green Rovers were enough for Town to progress to the quarter-finals of the competition for a second season running.
The Glovers were knocked out at the last eight stage by Luton Town last term and Way wants the players to live their ambitions and play in the final at the home of football.
“I’d like to go all the way this year,” he told iFollow Glovers.
“As a manager the competition has given me the opportunity to manage against Chelsea Under-21s and Bristol Rovers last season where we got to the quarter-finals.
“It was a good run and good for some of the players who weren’t involved as much.
“The group we’ve got are very ambitious and I’d like to see them run out at Wembley.”
He continued: “Forest Green pinned us in a little bit in the second half but the players showed stubbornness; a real resilience and it’s another clean sheet.
“There were a number of good performances and we could have scored two or three more in each half.
“Overall I’m pleased and it’ll be an unbelievable week if we can go up to Port Vale and do our best to turn around our league form.”