Officials at the Oxfordshire club's Alfredian Park announced Woodward's decision to stand down in a statement on their website naming Ian Glover as their new chairman for the remainder of the EVO-STIK League Southern's 2016-17 Division One South & West campaign.
Wishing Woodward well on Friday, the Alfredians said: "Wantage Town FC wishes Tony a speedy recovery and hopes to see him back involved in the running of the club in the future."
The second from bottom Alfredians host Wimbourne Town in league action on Saturday.
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