You can,if you take cuttings when the leaves are on they are called summer cuttings,there are whole topics in the forum for it.
My main way of acquiring varieties is to just walk about on my lunch breaks,I now have 7 unknown trees under cultivation and about the same number I'm yet to take cuttings from.
I just snip of a brace a at a logical point(just a light bit of pruning,not literally a branch),making sure there is some brown wood.Snip the tip off and the leaves,snip into pieces about 8" long,sloping cut above a node at the top,flat on the bottom,chuck them in the Baggie,label it and go about my business.
I am not suggesting you lean into someone's garden or anything,if it would require that sort of effort enquire and you may get some history as well,but if it's in public space the municipality/local government employees would probably chop it back at some point anyway and there's no harm in it.
The only issues with summer cuttings I see are;
1, you will have to over winter it indoors or in a greenhouse or something,if it roots and then goes dormant straight away it might not come back.
2,green wood does seem to dry out quicker if left exposed unless misted (I do mine in plastic mini greenhouse so hasn't been so much a problem).
3,for some reason success rates do seem to be higher with brown wood,whether dormant or not,I find green wood more prone to rotting if too wet,humid
4,green wood,if not potted up straight away does not survive as long in storage and success rates fall a lot if posted etc
I take summer cuttings mainly because
1,it's when I tend to find the fig trees,easily identifiable by the leaves
2,I can't wait to get growing them
3,provided I can keep them not to dry,not too wet,not too cold and not too hot then they work fine,a bit more Goldilocks than science in my experience.
If you want cuttings SENT to you from somewhere is would be best to wait until the trees are dormant,then you don't have to worry about things as much.The normal most viable and reliable method of rooting cuttings is with dormant wood,BUT summer cuttings do work