Hi, rsmith,
Take a look at the cathedral's compilation of statistics, you can see month by month for each camino to give you some idea. Of course that won't give you an idea of what those numbers mean on the ground, but you can see when the most popular times are to walk.
I've walked this route twice, once in late September, years ago, and there were always people at night in the albergues and opportunities to walk with others during the day if you wanted.
The second time was just a few years ago, and it was in late June, about the time I'm planning to go this year. It had a healthy group of pilgrims, never any crowding except at one point where there is a shortage of spaces (Berducedo), and beautiful weather.
I'm going back in late June this year, I hope, and I see that the numbers from June of last year are twice the numbers for the year I last walked it. So I expect there to be some crowding. But
@Juanma reports the opening of several new places along the way, so I don't think you should worry about a bed rush. Except perhaps in August.
The preference of month is such a personal thing, but I personally think the Salvador/Primitivo are at their peak in summer, so I'm hoping that late June is the sweet spot -- great time of year, and manageable crowds.
Buen camino, Laurie