One possibility is to set up a linked server on your local PC to the web site (which I assume is an SQL database at a hosting provider):
EXEC sp_addlinkedserver MYWEBSERVER, '', 'MSOLEDBSQL', 'mywebserver.hoster.com'
EXEC sp_addlinkedsrvlogin MYWEBSERVER, 'false', 'YOURMACHINE\USER', 'sqllogin' ´, 'password'
Now you can run things like:
INSERT MYWEBSERVER.db.dbo.tbl(....)
SELECT ...
FROM localtbl
At least in theory. Linked servers often mean hassle, not the least if you try to insert data in this direction. It works a little better if you insert into a local table, reading from a remote.
I don't know the specifics in your case, but I suspect that you will not be able to set up a linked server on the SQL Server instance for your web set.
An alternative is to use something that connects to both databases, for instance an SSIS package. The export/import data options in SSMS may also work for you.