Carmichael joined Bournemouth at the age of 10 and progressed through the youth system at the club.
In the summer of 2011, at the age of 16 he signed a two-year scholarship with the Cherries and made his Football League debut on 17 September 2011, in a 2–0 win over Exeter City. Josh the starting XI for the first time on 2 October 2012 in a 3-1 away defeat by Crawley Town.
He has also represented his nation (Scotland) at under 16s and under 19s level.
He joined The Boro' on a one month loan from his parent club in February 2014 and remained at Privett for the remainder of the season.
Josh rejoined The Borough after his realease from AFC Bournemouth in August 2016.