USB - C connector purely for charging a battery
Yes, Type-C connectors, in a standard way, do support currents up to 5A. In standard way the connector uses 4 contacts for ground, and 4 contacts for VBUS, making it at 1.25A per contact. So you need to make sure that your design (on PCB and on cable side) uses all (4+4) pins. You can use more contacts if you need more current, but I would advise against it, to avoid port destruction in case if someone decided to plug your cable into their laptop/phone.

However, if you plan on productizing your charger (and device), a simple two-wire power connection will be technically illegal from Type-C standard. In Type-C standard the VBUS must not be present on wires until the cable is plugged into a Type-C receptacle. To achieve this you will need at least one (thin) signal wire in your charger cable. This wire should be connected to one of CC pins and have a 10k pull-up to +5V. Your device receptacle, in turn, must have two 5.1k resistors to GND on each CC1 and CC2 pins of your receptacle.

Your charger side should sense the voltage level on CC wire, and turn VBUS on only when it senses the voltage drop on it (due to 5.1k pull down after connect). A simple analog comparator with right threshold and a high-side power switch (or just a power P-FET) will do the job. In this arrangement your charger (with Type-C cable) will be fully compliant and won't harm any other Type-C user devices.