The 20-year-old, who was released by the Foxes at the end of last season, is expected to go straight into the squad for Tuesday's game at Aldershot.
The deal will be funded by a group of 60 Exeter City fans who contribute monthly payments to the Exeter Exiles administered 1931 Fund.
"This is the 10th season in which City fans have been proud to boast that they own their football club," City chief executive Julian Tagg told the club's official website.
"This signing is further proof that we're all still pulling together to make our club a success.
"We all want Elliott to be a big success here, and hopefully being the 'fans' own player' will inspire him to do just that."