Consider the possibility that it sucks because there is no money for an update and/or it isn't a priority. Try to verify that this is a viable business and that an update would at least pay for itself, preferably more than pay for itself.
Consider the possibility that obvious need of an overhaul (in your eyes) is not very good evidence that these are good prospective clients. If you are offering charity, obvious need is a good metric for whom to target. But if you want a healthy business relationship, that is insufficient. A healthy relationship is symbiotic. It is one where both sides benefit. If they can't afford to pay you or just do not care, the suckiness of their site is not a good argument for viewing them as good people to target as future clients.
Don't hesitate to email them and pitch them. But don't be shocked if the success rate is low. You may need to come up with additional metrics to define "good prospective clients."