mosaicing them together isn't a problem since the tools are readily available. A problem that I have seen arise is that a gap can appear along the seam if there is not a slight overlap in the tile extents. This sliver effectively produces a nodata stripe making traverse impossible or worse... it becomes a preferential travel route. Filling gaps in dem files is well covered in many threads on GeoNet. One common option is to filter using a kernel wide enough to cover any gaps. Another is to produce a seam and use Con to use the values in the good portions of the dem and use a focal mean across the seam.
So, if you have nodata stripes, you have to deal with them or find a seamless data source.
Mosaicing doesn't appear to be the problem here, but if it is, then the options identified by Chris should work with no problem.
I don't know what "won't download to my map" means? Data normally has to be downloaded, perhaps unzipped, then added, so some clarification or confusion arises here... and hopefully this is being done in ArcMap and on in AGOL or someother environment