Когда-то была лиса бродячая из-за дров.
A couple notes about vocabulary: AFAIK, the verb бродить is nowadays pretty rare in the sense of "to walk/go slowly," even though that was its original meaning. (Believe it or not, nowadays the verb is more likely to mean "to ferment" or "to spread," as of yeast or mold! And when бродить is used with the meaning of walking, it tends to means something like "pace back and forth," like a caged animal.) So a much better translation for "to stroll" in this context is (по)гулять.

And дров nearly always means dead, dry "firewood" (while the related дерево can mean either "a living tree" or "wood" as a material that a carpenter uses). "Wood(s)" in the sense of "a forested area" is лес.

Finally, the standard cliche opening for fables and fairytales, equivalent to "Once upon a time," is жил-был (for masculine subjects, or жила-была, fem., or жили-были, pl.)

So I would translate this first sentence as:

Жила-была лиса. Раз она гуляла по лесу.
(There lived and there was a fox. Once she was strolling through the woods.)