The short answer is no; the longer answer is kind of.
I co-founded a pitch consulting agency targeting venture stage startups. Through the process of pursuing new clients I quickly learned which types of leads were worth the time and which were a dead end. Here's why VCs fall under both categories:
It's rare for a VC to spend their own money on resources for startups in their portfolio for a few reasons.
1.) Most VCs opt to take a largely hands-off approach. Their role is primarily as a funding source and sometimes a mentor.
2.) The VC has already invested money in the company. VCs commit money for equity so that the startup can then use that money on expenses such as web development. If a VC is going to spend money out of pocket, they will expect equity.
3.) Startups looking to give up equity in exchange for resources such as web development will pursue incubators and accelerators, rather than venture capital.
The longer answer is "kind of" because, assuming you're asking this question because you run a web development agency and are looking for clients, VCs are often a source of mentorship for the startups in their portfolio. If you are able to curate a relationship with a VC it is likely that when a startup in their portfolio mentions that they are in need of web development services the VC will recommend your agency. Having a mentor recommend your service is one of the smoothest ways to onboard a new client.
If you're looking for a more direct link to a portfolio of startups, I would recommend focusing on building relationships with incubators and accelerators. A successful sale will often mean that the incubator or accelerator pays for your services for dozens of their active startups. The best part is that accelerators and incubators are cyclical so once those startups leave the incubator or accelerator you can keep them as clients while also onboarding all of the new startups going through the program's next cycle.
If you're looking for additional guidance to make your agency a success feel free to schedule a call with me and I can answer all of your questions and help you lay out a plan to consistently reel in paying clients.