Yes, this is definitely a problem. I found the same problems back when I played WOW. Always looking for a tank or healer. I'm not sure why this is the case with league. Is it simply because to fill a team at least three (typically 4) people need to play damage dealing roles so there are naturally fewer support players in the pool? Or maybe a more likely reason is that being a support is simply less appealing? I do play support occasionally but not nearly as much as I play jungle or top. I think this is also the case for many other players because playing top,mid,jung, or adc is more satisfying because you get to kill, gank, and pwn the other team while the supp is just there to stop this from happening to his/her own team. I am not trying to belittle a supp's role at all, they are a critical part of a team and a having a good or bad support can make or break a game. However I think the problem is that the glory and attention almost always goes to the champ that got the most kills or objectives and rarely to the champ who saved some guy from certain death countless times or sacrificed himself to save his team from a bad engage. I think if the community/Riot can work to make the role of support more satisfying or visibly important then this problem of long waits for a supp may go away. One possible solution is to somehow track near death saves or something along those lines. I know that is not necessarily something that is easy to track but I think that if there was someway of showing how a support performed that would be a start in the right direction. Maybe something that showed "lives saved" or something like that on the main post game screen could be a solution? In a nutshell, the problem I see is that all the glory goes to the the killers and none to the life savers.
I know that at times other roles are hard to come by, but the majority of the time it comes down to "dang, where is this support? I've been waiting 10 mins..." situations. Sorry this is really long winded, I only intended to say a couple sentences lol.