Ian, 52, said this election would mark the first time he was unable to vote since 1992.
Knowing he would be on holiday on the date of the Senedd election, Ian said he sorted out the paperwork in advance but became worried when his postal vote did not arrive.
He said he received no communication from the council about delays.
“My democratic right to vote has been taken away from me by a quite obviously ineffective process,” he said.
“I am also disappointed and angry that Cardiff Council have not communicated any delays. I have found out everything via social media.”
Cardiff Council said it had has issued almost 47,000 postal vote packs and more than 26,500 had already been returned.
“We have been working urgently to investigate the issue and have identified that the affected packs relate to a supplementary print run for the Caerdydd Ffynnon Taf and Caerdydd Penarth constituencies,” it said.
“We have received assurances from our print provider, Civica, that the relevant data was received correctly from the council and that the packs were printed and dispatched to Royal Mail, who continue to investigate.”
The council added that “a number” of postal vote packs had been reissued and further replacements would be hand-delivered on Tuesday, with “a small number” sent out again by mail.
“Voters who receive a replacement pack can complete it and either return it to our staff at the time of delivery, hand it in at any polling station before 10pm on Thursday, or deliver it to County Hall before 10pm on polling day.
“We regret any inconvenience experienced and are focused on ensuring all those affected are able to vote.”