Run a help desk — Voicemail
Give the desk a phone number and every message left on it becomes a ticket, with the recording to play and a transcript to read. Callers hear your greeting first; leave it empty and they hear your business's name.
The number belongs to your desk alone, so a call to it can only land in your table. The ticket is created the moment the caller hangs up, which starts the response-time clock and assigns somebody straight away. The recording and the transcript follow a minute or so later.
To keep the number your customers already call, ask your phone provider to turn on conditional call forwarding (forward when busy or unanswered) and send those calls to your desk's number. Your phone still rings first, and only a call you miss becomes a ticket.