Leicester's relegation to the Championship was confirmed after a 1-0 loss to Liverpool, their 23rd league defeat of the season.
The team struggled to score at home, with nine consecutive matches without a goal, a new English record.
Vardy, who led the team to a Premier League title in 2016, is their top scorer with seven goals, but has gone 10 games without scoring.
Vardy described the season as a "total embarrassment" and admitted that the team "failed" as a collective.
"I'm at a loss for words," Vardy said. "There's no excuse for our performance this season. We've let ourselves and our fans down."
"As players and as a club, we've had many highs and successes, but this season has been miserable, and personally, it's been a huge disappointment. It hurts, and I know it hurts you too."
"To our fans, I'm sorry. Sorry we haven't performed, and sorry for the way the season has ended."
Vardy, 38, is in his 13th season with Leicester and his contract is set to expire in the summer.
Manager Ruud van Nistelrooy is also under scrutiny, having taken only four points from his last 18 games.
Leicester has five games remaining, starting with a match against Wolves on Saturday.