Former Designer here, turned Dev, and I used to piss and moan about Web Controls too. Honestly, its MUCH cheaper for a designer to adjust their practices than for a .NET Developer to delve into a custom impelmentation of a GridView because the designer INSISTED that each TD have a 'rel' tag (or whatever).
As MainMa very wisely pointed out, the decision to use Webforms is a company choice that limits some of the control over the HTML while granting some efficiencies in coding, plain and simple. Unless the company is willing to re-consider (which they should NOT do just to please the designers), then the designers need to accept this reality. Here's a few things they can do:
1) Stop depending on ID's for anything. Even though this felt wrong at first, I found that life was actually a lot easier when I styled everything with classes (and inheritance, of course). First of all, it equalized all my selector weights. In CSS inheritance, ID trumps CLASS. It was actually kinda nice to have everything be a child and/or class selector, and made figuring out the specificity order a little simpler. Same thing in the JS layer, it gave me ZERO pain to swap out my ID-based selectors for class-based ones.
2) Teach them what RadioButtonLists and CheckboxLists convert into, along with Label = span, Panel = div and the other non-obvious control-to-html stuff. The way .NET renders these to HTML was a little weirder than I expected, and it was much easier for me to create screens when I knew how the HTML would come out of those controls.
3) Have them do their designers IN ASPX DIRECTLY, not raw HTML (!important) . Teach the designers the basics of GridViews, ListViews, etc. Give them some code snippets to push an anonymous object collection into a Grid/ListView control. If they can learn CSS then they can learn to copy-paste this code in. They can use the free version of VS Web Express, which is quite good at CSS & JS work now. These dummy web projects will give the designers a chance to enter some controls, then View Source to see how they are rendered.
4) (update) Explain how the FORM tag is used in .NET. Forgot about this one earlier, but another thing the designer has to get used to is that usually, a single FORM tag wraps the entire page. This alters the way form controls behave, and you can't nest FORM tags without really weird side effects. Make sure the designers understand this or else their form HTML will be a nightmare to make into WebForms.
5) (update) Stay away from Themes and Skin. Even though the .NET framework has these tools to help style controls across an application, they are clunky and weird to normal Web Designers, and I never found them worth my time. They seem like a nice tool to devs who aren't well versed in CSS, but will only slow down the designers. Let the designers work in their natural environment (html & css files) and they will be happier and more productive.
6) (update) Keep "prototype" projects in your site solutions. To make sure the developers always have a target to code against, have the designers create a fake web project in your real solution to keep their ASPX-only pages preserved and untouched by the real developers. This means the designers can look back to their prototypes in the same solution as the real project to check on how the developers did, and the devs can run the prototype at any time to make sure their work matches the designers' intent.
Finally, resist any complaints to convert to MVC, unless you are ready to re-train your Devs. I love MVC personally, but if you've got a team with a ton of WebForms knowledge, don't go throwing that away for no reason. If your applications are having ViewState problems, SEO problems, or accessibility issues, then absolutely give MVC a hard look. But it will take a LOT more time to train WebForms devs on MVC than it will to train Designers how to use Web Controls.
At the end of the day, there was NO DESIGN I CAME ACROSS, that I couldn't personally make work in WebForms, even if I ended up swearing at that darn GridView for an hour before figuring it out.
Are there any tools or applications to provide the designers with the
rendered (html page) of the .aspx pages?
[10/10/12 - added points 4, 5 & 6, added comment about tools/Expression/VS]