Absolutely! If you did the work you were expected to do, and there was an agreement in place beforehand (even verbal), then I'd expect you to send them a bill.
This client may decide not to pay it, given that "you didn't fix it yourself", but you still need to show that your time is valuable! Hopefully, lesson learned.
In a previous job, when I was working for a tech shop, we would have people trying to come in for free advice all the time! I'd offer one or two sentences for a solution that people in the know would understand, and if they were asking for more details, I'd explain our hourly rate. Some people wouldn't bat an eye at the cost, others would get defensive and try to work around it. Stand your ground, and stick to your guns - you didn't learn all your skills for free, just to give it out for free to everyone who asks, right?
In the future, I would explain with the quote that you expect X Y and Z conditions to be met, in order to facilitate you completing your job as close to what the quote says, as possible. If you need to overhaul a web site, explain you will need Admin access to the hosting service to do your job. You are free to explain why, but you need to know what you need to get the job done, as the professional.
When I do web site consulting, I know my limits, and I ensure the client understands this (i.e. I don't do any design work at all - check my blog, it's just the default theme). I can do all the server back-end stuff, and run quite a few sites this way, and tell the client they need to hire a professional to do certain things that I can't, and I encourage them to have the professional contact me. I allow them supervised access, or I ask for the files to be FTP'd to me directly, then I'll post it. As the Server Admin, it allows me to do backups and cover my ass in case the other professional messes something up, in particular.
If you are doing lots of free work, the client is going to expect you to keep doing free work, even outside the scope of what you can handle. Stick to your guns, charge after a certain amount of time, or else explain why you can't continue as previously planned.