No, in is the appropriate preposition.
Membership or participation in a group who are communicating with each other is expressed by in.
Once you're in our network, we will direct potential customers to you.
That is, once you join or become a part of our network...
We would use on to refer to the communications mechanism or platform itself:
I haven't been getting any new patients since joining your health-care
--Are you sure you're properly registered on Health-Link, our membership portal?
Company-names can also play on associations or connotations.
To be "clued in", for example, means to have needed information, to be cognizant of what one should know. To be a part of the "in-crowd" means to be belong to a group that many people wish to be belong to. The word in connotes certain benefits.
P.S. It is possible to understand a company such as Linked In as a communications platform or as a network of members. Depending on how the speaker is thinking of it at the moment, the preposition could change.