The short answer is, yes, if you want them to. By the time the shiny world of 2019 rolls around, base-level Greentec (hand-wavy "don't worry about batteries" technology) smartphones would be essentially free/ubiquitous, depending on how much of your consumer soul you were willing to sign over in the form of advertising/data-mining rights.
So if you want to participate in the Forums whenever, or even look something up on the site In-Game (if you actually have your own internet-capable phone), feel free - this is not something you need to spend Gear levels on - it's meant to enrich the game, and it's more enriching the more people use it. But note that a "free" device can do *nothing meaningful* (including "work as a phone") other than access Time's Up - if you want it to have In-Game usefulness, it's Gear (or possibly Gadget).
If for character/story reasons you prefer *not* to have such a device, you still might run across one in your wanderings that worked for a little while, or was part of a fixed installation (a Greentec info kiosk at an interstate rest area, a dusty and battered terminal at a public library that happened to install flexible solar panels that survived Collapse, etc.) - so don't let that keep you from reading the site or posting in the forums.
That's covered in about as much detail as we intend to in the "HOW THIS SHIT WORKS" section of the main Time's Up page. If you're not sure if you can/can't do something after reading this, ask on the main Source list (or if you must, email privately).
If you have high levels of Technical Science and want to fuck around with PNet or Time's Up, email privately - but don't expect much - for the moment, TU is an important IG resource and has significant Plot Shielding.
Please don't. Aside from: this page, the link to it on the front page, the reality link at the top, the copyright notice, and very infrequent "Forsooth"'s only when necessary, we want Time's Up to be *entirely* IG. If you have an OOG question or want to make an OOG aside about an IG post, send mail to the source list (firstname.lastname@example.org) and add a link to the forum post for context if you need to.
Spammers are fairly good at getting past registration obstacles, so I had to add an OOG validation to the registration form to prevent all the spam registrations I was getting. The password is available on the main mailing list, look at the archives or ask (and be mocked for not looking at the archives).