Gus Poyet's men capped off their campaign with a 3-1 home defeat to Swansea City on Sunday after making sure of their Premier League status by beating West Brom 2-0 four days earlier.
The Stadium of Light outfit have six players whose contracts are set to expire while a number of loan signings are poised to return to their parent clubs.
But Wickham, whose goals proved pivotal in Sunderland's run of four wins in their last six games that secured survival, insists the players that do leave can do so with pride.
"The club can do their bit this summer and I'm sure they'll be busy," Wickham, who had spells on loan at Sheffield Wednesday and Leeds United, told The Sunderland Echo.
"But the lads can be proud of what we've achieved.
"Even those who are out of contract or on loan, I'm sure none of them wanted to get relegated. We all dug in.
"A few of them will go and we'll be left with a bare squad. But at least we didn't get relegated."
Wickham was disappointed to finish the campaign on a losing note, but is confident that Sunderland can improve on their 14th-place finish next term.
"It was disappointing to end with a defeat, but I think we can take spirit from what we've achieved this season," Wickham added.
"Not so much finishing where we have, but, considering the circumstances we were in four weeks ago, I think we can hold our heads high and be proud of what we've done.
"We haven't ended on a high, but we can enjoy the summer now, come back in July and work harder to be better next season."
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