And AFAIK the answer is no, they don't. Running a toll without a transponder is an offence, even if they don't have a cash option. For tollways (or exits) without a cash option, don't get on the tollway. They don't have a system like 407 where they mail out bills to motorists. If you know you have accidentally run a toll, go online and pay the appropriate toll to and I doubt you will be fined (speculating). Also speculating that unpaid tolls stick around like parking violations; so if you are pulled over for something else, your unpaid tolls could rear their ugly heads.
Curious why you wouldn't buy a transponder for 1-2 trips a year. I have them in both our vehicles, and we only go down to FL every couple of years. The transponder savings in one return trip on the 417 tollway around Orlando pays for the transponder itself. If you blow a toll on your first trip, your plate will be on file potentially forever, making future trips risky.
ETA: getting a transponder couldn't be simpler. You can order one online and they will ship to Canada for free. Only takes a few days to arrive and can be installed and ready to go before you leave.