Here is an adaptation of a couple of my answers to similar questions.
Converting automatically is a better choice than rewriting. It's a common pitfall to start out optimistically rewriting a large piece software, make good early progress fixing some of the well-known flaws in the old architecture, and then get bogged down in the functionality that you've just been taking for granted for years. At this point your management begin to get twitchy and everything can get very uncomfortable.
...and here's a blog post by a Microsofty that agrees with me:
Many companies I worked with in the early days of .NET looked first at rewriting driven in part by a strong desire to improve the underlying architecture and code structures at the same time as they moved to .NET. Unfortunately many of those projects ran into difficulty and several were never completed. The problem they were trying to solve was too large
This excellent Microsoft page recommends two third party migration tools as better than the built-in VB.NET upgrade wizard - Artinsoft and CodeArchitects VBMigration. I believe they have some support for common third-party controls and DLLs - Artinsoft suport these. It would be worth contacting them with a list of your dependencies. VBMigration has a free tool that will list the dependencies for just this reason. Also worth contacting the original vendors in the hope of a .NET equivalent.
The Microsoft page also says:
Performing a complete rewrite to .NET is far more costly and difficult to do well [than converting] ... we would only recommend this approach for a small number of situations.
There are many more C# developers than VB.NET developers on Stack Overflow, so you will probably get several answers recommending C#. Also historically Microsoft has tended to support C# more enthusiastically in terms of code examples for new parts of .NET and so on. But Microsoft does now assure us that:
Both [C# and VB.NET] are first-class programming languages that are based on the Microsoft .NET Framework, and they are equally powerful.
So it's a personal decision whether you want to choose C# or VB.NET. The Artinsoft tool claims it can convert VB6 to C#.
EDIT: I just found another offering - NewCode - through an ad on a programming website!
My snap judgement is that the website isn't as detailed as the two competitors I've written about above. That may be unfair. Some guy at Microsoft Ireland blogged about them - I think they are based in Ireland. Apparently the tool converts your VB6 to a DSL and then to VB.NET Winforms, C#, WPF, Java...