How Should I Reach Out To SOLIDWORKS For Technical Support?

To give the short answer at the top, you should reach out to your SOLIDWORKS Reseller first before attempting to contact SOLIDWORKS directly, for almost any reason. If your VAR is CADimensions, you can put in a technical support ticket here:
https://www.cadimensions.com/contact/support/
or call us at:
(877)223-4255

To elaborate a bit more, SOLIDWORKS almost never works with customers directly. Instead, they work through Value-Added Resellers, or VARs for short. Companies like CADimensions that work directly with customers on SOLIDWORKS's behalf. That way, if you run into a circumstance that requires you to communicate with SOLIDWORKS, you can reach out to a local organization first. You'll work with people that know your language, customs and culture, rather than someone half a world away who's struggling to understand your circumstances.

We VARs are here to help ensure you get what software you need, get you up and running, answer questions you may have, and resolve any technical issues that might emerge. The other side of that coin, however, is that SOLIDWORKS is expecting you to reach out to your VAR first, and so if you do contact SOLIDWORKS directly, there often isn't a ton they can help with. A lot of the time they will just tell you to go talk to your VAR about it.

As such, there technically is a mechanism for you to put in a tech support request with SOLIDWORKS directly. But even then, it will not directly reach out to someone at SOLIDWORKS, it will just become available to someone working on the support team at your VAR. They are expected to try and resolve the issue as best they can before ever getting SOLIDWORKS involved. And if they cannot resolve it on their own, they need to collect as much data as they possibly can about the problem before they extend the problem to SOLIDWORKS themself.

However, for CADimensions, and probably for most other VARs, this process is a bit cumbersome. It sits outside of our normal support ticket infrastructure and so we usually have to log in, close the ticket you created, and manually create our own ticket within the system we actually use. And if the problem actually does need to be elevated to SOLIDWORKS, we would probably just create a new ticket with them ourselves instead of continuing the one you made.

In general, trying to contact SOLIDWORKS directly instead of reaching out to your VAR usually just makes the process more complicated for all parties involved, while also making it take longer. 

So, how do you find out who your VAR is if you don't know offhand?
Whomever sold you the seats of SOLIDWORKS you're using is most likely your VAR.That info can be found in the purchase orders, as well as any email communications involved in that process. If someone has recently resolved a tech support issue with your license of SOLIDWORKS, they either are your VAR or know who your VAR probably is.

If your VAR happens to be CADimensions, you are welcome to contact us a variety of ways.

If SOLIDWORKS isn't working, or you otherwise need technical support, please put in a ticket here:
https://www.cadimensions.com/contact/support/

If you would like to call us, you can reach us at:
(877)223-4255

If you still want to contact SOLIDWORKS directly, there are a few ways to accomplish that too.

Their phone number is 1-800-693-9000 if you're calling from within the US and Canada or 1-781-810-5011 if you're calling from outside the US and Canada

They have a number of resources available here:
https://www.solidworks.com/contact-us

including a number of useful, more technical resources here
https://www.3ds.com/support/cloud

If you absolutely must put in a request to SOLIDWORKS directly, even though it will still go to your VAR first and most likely just complicate matters, you can click the Submit a Request button on the 3ds.com/support/cloud page above, and it will take you there.