Thank you for trying out LeoInteg! It's with report like yours that I can make the whole process work on more platforms for the most people possible! :)
Is it your first time trying to use LeoInteg?or was it working for you before and it somehow stopped working recently?
Lets see if we can find more info somehow to help me find a source of this failure... :
Can you paste the output of your server starting? And maybe the output of vscode->help-> about.
Also, can you paste a screenshot of your LeoInteg options? Such as this for example:

It may help to uncheck 'start server' and 'Connect to server' to control when you try to start and connect, just to go step by step to see if something can be spotted that blocks the connection somehow.
Also, you may try (start server off) to start it yourself from your /leo/core folder. The script is leoserver.py. Then, back in vscode, press the connect button. (instead of pressing start server, as it's already started.)
Lastly, are you using leoInteg 1.0.7? (you can verify in the "Leo Welcome" view shown above.)
I'll post some more suggestions if I think of more things to look at.... in the meantime go ahead and try out some more while gathering some more details like I suggested
And again: Thank you very much for this report!
Félix