Town were first to threaten when Wickham's long throw dropped to Grant Leadbitter 20-yards out but he dragged his shot wide, while Doncaster at least tested the goalkeeper on 13 minutes when captain Gareth Roberts' 25-yard drive drew a diving save from Brian Murphy.
James Chambers got in front of Shane O'Connor but his cross was just too strong for the lurking Paul Heffernan before Leadbitter volleyed across goal after his corner had been cleared back to him.
Jordon Mutch fired a snap shot wide from 30 yards before an offside-looking Tommy Smith headed a Leadbitter free-kick back across goal for Wickham to turn and hook his shot over the top.
Soon after, Smith saw a header drift wide from an O'Connor cross, Wickham fired wide from a Jack Colback pass and James Hayter curled wide after Heffernan laid off a Roberts cross neatly.
The second half started equally slowly with David Norris being denied by Chambers as he chased a Wickham pass before Wickham tested Neil Sullivan with a header from Colback's cross just after the hour.
Murphy cleared a John Oster pull-back before it could get into the danger zone and Hayter nodded wide after climbing above Murphy as they contested a Jay Emmanuel-Thomas cross.
Ipswich broke the deadlock on 72 minutes with only their second effort on target and it owed much to a defensive mix-up as Wickham pounced for his fifth goal of the season and third in the Championship.
Gareth McAuley's ball forward saw Jon Walters compete with Roberts, and although the defender got to the ball first, his challenge sent the ball square across his own goal and Wickham beat Shelton Martis and Sullivan to the loose ball to clip into an empty net.
Ipswich were buoyed by that goal and they could have doubled their lead when Walters collected a Leadbitter pass and cut in off the left flank before firing over the top.
Out of the blue, Doncaster levelled on 82 minutes when O'Connor misjudged a crossfield ball from Roberts, with his header dropping behind himself, and as he tried to get to the loose ball, he nudged over Shiels for an obvious penalty, from which Shiels sent Murphy the wrong way.
Ipswich felt they could have had a penalty when Sullivan seemed to leave his foot in on Wickham and the frustrated teenager was then cautioned for a lunging challenge on Roberts.
Both teams had chances to win it late on with a Chambers cross-cum-shot being tipped over by Murphy while Leadbitter volleyed wide from distance for Town, but a draw was just about the right result.