Brendan Rodgers' side have now gone 17 games without a win at the Madejski Stadium - a sequence which goes back to late January - but it was only a combination of poor finishing and bad luck which denied them victory.
Both sides remain in the relegation zone - Reading are 22nd, two points and a place above Ipswich - and will need to turn things around quickly if they are to pull away from danger.
Ipswich captain Jon Walters shot over from long range before Roy Keane's visitors took the lead in the ninth minute.
Damien Delaney hoisted a long throw into the penalty area and the Royals made a hash of clearing their lines. Ivar Ingimarsson got his head on the ball but could only find Stead in space and the former Blackburn man scuffed a shot which beat Adam Federici and found the back of the net.
Gylfi Sigurdsson then saw a shot deflected narrowly wide as Reading pushed for a response and fellow midfielder Brian Howard went even closer moments later when he ran forward and crashed a shot against the crossbar from the edge of the area.
Ipswich goalkeeper Asmir Begovic almost gave away a penalty as he raced out to challenge Church, but clearly got a toe on the ball to deny the Reading man a shooting chance.
Tamas Priskin almost doubled the visitors' lead against the run of play with a fabulous effort from 25 yards but Federici recovered to make a fine save - and Grzegorz Rasiak should have equalised moments later.
Jobi McAnuff tricked his way to the byline and crossed to Rasiak but the Poland international headed inches wide from inside the six-yard box.
It was a poor effort from Rasiak but he made some amends by playing his part in the 47th-minute equaliser from Church.
Rasiak dinked a cross into the area and, after Alex Pearce won the header, Sigurdsson lashed a shot across the face of goal where Church was on hand to divert the ball home.
The goal was deserved but Delaney almost put Ipswich back ahead with a flicked header from a Carlos Edwards corner but Federici made a great save.
Shane Long saw a header hacked off the line by an Ipswich defender as the hosts piled forward in search of a winner and Church was desperately unlucky not to find it with 15 minutes remaining.
Substitute Jimmy Kebe crossed for Church to hammer a shot against the underside of the crossbar, and McAnuff's header from the rebound was well cleared by Alex Bruce.
Rodgers felt his side should have had a penalty in the closing stages when Long was brought down in the area but both managers were forced to settle for an unsatisfactory point.