No financial details of the deals were disclosed, but British media reported that Vokes was valued at around 9 million pounds.
Crouch was part of the Stoke team relegated to the second-tier Championship last season but has not started a game since September.
The former Tottenham Hotspur and Liverpool forward, whose deal with Burnley runs until the end of the season, has previously played for six Premier League clubs in his career and scored 108 top-flight goals.
He also won 42 caps for England, scoring 22 goals between 2005 and 2010.
Despite his age, Crouch said he still had the ambition to play at the highest level.
"I still want to achieve things," he told Burnley's website. "I want to get more games and I would love to add to the 100 Premier League goals.
"My game has probably changed a little bit. I'm not 25. But my attributes haven't waned. I never really had blistering pace, so I haven't lost that. I know I can impact games at this level."
Vokes, who waved goodbye to Burnley fans after the 2-2 draw with Manchester United at Old Trafford on Tuesday, has signed a 3-1/2-year contract at Stoke, having struggled to secure a regular starting place of late at Turf Moor.
The Wales international won two promotions to the Premier League with Burnley in his seven seasons with the club, scoring 62 goals in 258 appearances.
"It's great to have it finally done," he told Stoke's official website. "It has been a long day and a lot of waiting around but I am delighted - I just can't wait to get started now."
The 29-year-old joined Burnley from Wolverhampton Wanderers and was part of the Wales team which reached the semi-finals of Euro 2016.
"Burnley was a huge part of my life, I really found my home there and I made a lot of good friends there, but it was time for me to move on - and what better club to do that with than Stoke City," Vokes added.
(Reporting by Simon Evans and Simon Jennings; Editing by Toby Davis)
