The son of former Norwich City manager Peter Grant and brother of Raymond Grant, Peter Grant Jr was at Norwich City briefly as a youngster also playing for Thorpe Rovers.

After his father's departure from Norwich, he moved to Cambridge United's youth team in 2009/10. The following season, he had a trial with Watford before joining Peterborough United where he captained the youth team and signed a professional contract in 2013. He was loaned to St Neots and Thurrock in 2011/12 and to Histon at the start of the 2013/14 season.

Peter was released by Peterborough at the end of the 2013/14 season.

Peter signed for Scottish side Falkirk in June 2014.

Peter scored in the Scottish Cup Final on 30th May 2015 but Falkirk finished up losing 2-1 to Inverness Caledonian Thistle.

Peter was released by Falkirk at the end of the 2017/18 season having made 97 appearances, scoring seven goals. He reportedly then signed for Greenock Morton but instead signed for Plymouth on 27th June 2018.

On 29th January 2019, Grant became one of three players to join Carlisle (along with loan signings Nathan Thomas and Callum O'Hare). Grant terminated his contract with Plymouth to make the move, signing a short term contract. He had made nine appearances for Argyle. Grant was released by Carlisle at the end of the 2018/19 season. He had made 4 appearances for Carlisle.

Grant joined Greenock Morton on 8th July 2019.

On 24th January 2020 he moved to Queens Park. Head Coach Ray McKinnon said: “I’m delighted that Peter has agreed to join us. He is an experienced centre half who has racked up a lot of games at Championship level, as well as having played in down south despite only being 25. Peter’s experience and leadership will be a welcome addition to the squad and it’s a signing that I’m very happy to have been able to make.”

Queen's Park were promoted to Scottish League One as champions after a goalless draw with Elgin on 20th April 2021.

He signed a one year contract with Clyde on 7th July 2022 where his brother Ray had joined two months earlier.

In January 2023 he joined Dumbarton on loan.