I'm looking for a front thru axle skewer for my bike. The stock axle has these measurements, L124mm TL16mm TP M12 x 1.5. The closest I've seen so far online is L125mm TL17mm TP M12 x 1.5. Would it be okay to fit the one I've seen online as is, or should I opt out getting a washer/spacer?
Yes, in most cases you can do what you propose and it's fine.
To look cool, some frames have covers on the drive side of the axle, such that nothing can stick out unless you remove the cover. Canyon is one company that has this.
It's also pretty common for the stock axle to have more threaded length (TL) than it necessarily needs. As long as the axle you choose isn't running out of threaded length before it engages all the dropout threads, you don't need any more than that.