Location: The location of Mount Horeb, which most understand to be the same as Mount Sinai (see Deut 4:10, 4:15, etc), is unclear. There are many traditions about the location of Mount Sinai, some of which are probably more hospitable than others. The location offered by Open Bible is but one of many.
Sheep: None of the candidate locations appears to be especially fertile, but sheep can graze in arid land -- just not as densely or as effectively as on good land with ample vegetataion. Sheep spend hours per day grazing, so if they have to walk a bit to do that, that's fine.
Addendum: I mentioned this question to my husband, who has been to the Sinai wilderness (many years ago). He told me that while the mountaintops are arid, the places between them collect what water there is and so do better. His tour guide (whose agricultural credentials are unknown) told him that sheep do fine there. Of course, this is today, not then.
Going to meet God: Exodus 3:1 says:
וּמֹשֶׁה, הָיָה רֹעֶה אֶת-צֹאן יִתְרוֹ חֹתְנוֹ--כֹּהֵן מִדְיָן; וַיִּנְהַג אֶת-הַצֹּאן אַחַר הַמִּדְבָּר, וַיָּבֹא אֶל-הַר הָאֱלֹהִים חֹרֵבָה.
Now Moses was keeping the flock of Jethro his father-in-law, the priest of Midian; and he led the flock to the farthest end of the wilderness, and came to the mountain of God, unto Horeb.
The text says he "came to", not a stronger word like halak (went to, walked to). This is consistent with an understanding that Moshe was just minding the sheep, an inherently migratory activity, and happened to come to this place.
If Moshe had been seeking God, we would have to ask how he knew to look there in particular. Further, Moshe seems a little disconcerted by the encounter, hiding his face and initially declining the divine charter; that could still happen if he were seeking God (maybe God was way more overwhelming than Moshe imagined), but I think the simplest interpretation is most plausible: God called out to Moshe and, prior to that, Moshe wasn't seeking a particular place.
Finally, Google Maps is remarkably short of visual data from 3300 years ago, so it is also possible that even if this is the right location, conditions have changed.