Despite being the highest point in the continent and relatively remote, Mount Kilimanjaro apparently has good telephone and data service coverage, even at the peak.
Considering most carriers charge quite a bit for international roaming, your best bet would be to pick up a SIM card locally to swap into your phone. There are apparently a number of locations where you can buy and register a SIM card from retailers and re-sellers, with international visitors required to show their passport.
Some of the carriers reported to work on the mountain and surrounding parks include Vodacom, Airtel, and Halotel, each with 2G and 3G service, and 4G/LTE to a lesser extent. The SIM card itself typically costs on the order of TSH 1000 ($0.45), with 1 GB of data over one week of access costing TSH 5000 ($2.25). The SIM cards may be offered at the standard size, which few phones accept these days, but may be pre-cut to easily remove the extra material or manually trimmed down on location.
Be sure to obtain the text activation codes for receiving phone service and buying or applying credit. Additionally to ensure you can get connected to the network for Internet access, be sure to obtain the APN (Access Point Name) configuration settings.
There should be 2G and 3G service at a number of the camps, and the guides and porters typically know where to find service otherwise. With ascents taking around a week, it would be a good idea to bring along a high capacity power bank to ensure you have a charged phone to use.
References:
Stay Connected on Mount Kilimanjaro: Telephone and Internet Access