Short approach to get on a glacier in Iceland?

I'll be heading to Iceland in September with my wife for our honeymoon, and we were hoping to get on a glacier or find a hike that involves some mild glacier travel (I'd prefer not to bring ice axes, but I'm fine with packing crampons).

Anyone have any suggestions? We're on a tight budget so guide services are out of the questions (if it wasn't so expensive we'd be ice climbing!).

The wisdom of glacier travel sans ice axe notwithstanding, my wife and I were easily able to "scramble" onto Vatnijokull (that's a glacier) from the day-hike tourist trail in Skaftafell National Park.

Please refrain from accessing the glaciers directly off the tourist trails. This inevitably leads others to follow you. The glaciers have been retreating very rapidly recently and their terminal lagoons are in turn growing very large. I can't confirm the Skaftafell glacier tongue but f.x. the Svínafellsjökull glacier tongue is no longer accessible from the old point. Same applies to the Sólheimajökull glacier. Also note that if you are going onto a glacier where tours are being run you are not only endangering yourself but also all the other clients on the ice as the guides running those tours are your nearest SAR.

Stay safe :D

Source: Work as a glacier guide for Icelandic Mountain Guides.