I've noticed a few example of Torah observing Christians saying they don't go on The Internet on The Sabbath. I find that strange.
Any argument that going on the Internet qualifies as servile work would be as absurd as the Rabbis saying it would violate the Sabbath to push the button on an Elevator.
The Sabbath is also a day of Assembly, part of it's purpose is for God's People to come together. In the modern world The Internet is a vital part of how we do that. If you know other believers in your area you should feel free to consider communing with them more important.
But we should never neglect using part of The Sabbath Day to check whatever online groups and communities we are members of. And so on. Cause some of us lack local like minded Believers we can fellowship with in person.