Here's the short answer: No team can add Brendan Sorsby in 2026 — not the Jets, not anyone.
The NFL declined to hold a 2026 Supplemental Draft, which was Sorsby's only pathway into the league this season. He filed his application just three business days before the deadline with insufficient documentation, and the league also questioned his willingness to accept responsibility for NCAA gambling violations. Without the supplemental draft, there's no mechanism for any NFL team to sign him for this year (from the Brendan Sorsby page).
Your Jets are set at QB anyway — they've got Geno Smith as the locked-in starter, with OC Frank Reich raving about his "elite" football mind at minicamp, plus Bailey Zappe, rookie Cade Klubnik, and Brady Cook competing for backup spots (from the New York Jets and Bailey Zappe pages). So Sorsby wasn't a need even if the league had held the draft. His earliest NFL opportunity is now the 2027 draft.