Born in Stratford, Grotier started playing as a goalkeeper at primary school but his early career was curtailed once he moved on to West Ham Tech, with the presence of the West Ham Schools goalkeeper restricting him to the role of understudy.
He was spotted by Clapton and played in various Sunday leagues in the local area, before being approached by West Ham United chief scout Wally St Pier. After playing a game for the Hammers in the South East Counties League early in the 1966–67 season, he signed forms for a year-long apprenticeship on the day of his 16th birthday. After signing professional forms a year later, he was part of the West Ham A team which, as well as winning trophies in the Metropolitan League, won the Southern Junior Floodlit Cup in 1967 and again in 1969, when they also won the South East Counties League Cup.
Grotier made his first team debut for West Ham on 19 April 1969, a First Division encounter against Tottenham Hotspur. The game was the culmination of an eight-day period in which he also played for the youth team and the 'A' team, as well as two reserve games.
He played for the club until 1974, mostly as understudy to Bobby Ferguson. He went loan to Cardiff City in 1963, and then spent a month at Lincoln City at the start of the 1974–75 season. West Ham were asking for at least £20,000 for his transfer and his stay at Lincoln was only made permanent after fans raised money for the fee.
Grotier made a total of 263 League and cup appearances for Lincoln, including a run of 134 consecutive appearances, and was an ever-present during the 1975–76 season. His first two seasons at the club earned him spots in the PFA Division Four Team of the Year in both seasons.