Each NWS Weather Forecast Office (WFO) has their preferred methods and ways of doing things. Having said that, almost every WCM (Warning Coordination Meteorologist - the person who manages the Skywarn program at each WFO in addition to other duties) will tell you that the important thing is that you report regardless of how you do it. Here are common ways reports are made:
Ham Radio - One office that is really big into this method is Fort Worth (which handles most of north and even some of central Texas). They have a team of volunteers from WX5FWD that come into the office and sit directly across from the forecasters and the WX5FWD team communicate directly with nets in impacted counties. Each county has a repeater they use and those nets are managed by a local group in a given county (see
CWA Local SKYWARN Frequencies Map | WX5FWD SKYWARN® Team). These are typically ARES (Amateur Radio Emergency Service) or RACES (Radio Amateur Civil Emergency Service groups. Most (but not all) ARES groups will allow any ham to make reports that meet a given criteria and report style. RACES nets are restricted and only hams appointed by local governments can participate in the net except in some very limited exceptions. Other WFOs have different ham setups. Kansas is one big linked statewide net. Some offices don't use ham radio. Typically reports are made to a net control station who then relays the message to the NWS WFO either via radio or NWS Chat. The big problem for a chaser is 1) finding the correct frequency (although hamchaser.org can help out) and 2) having your report be accepted by the local group that runs the net.
Spotter network - This is the way most chasers report. It is quick, easy, and goes straight to NWS Chat. Some WFOs use NWS Chat a lot and other not so much.
Phone - All NWS offices have a phone number you can call to report. Finding the correct number can be hard. The office number is typically not answered, but is often a machine that answers and the forecasters can hear your report as you leave a message.
Social Media - In the past, WCMs would tell you to never report an event in real time via social media, as no one was watching it in real time. That has changed in recent years. Many offices either try to have extra staff or volunteers on hand to check social media directed at them or use what are called a VOST (Virtual Operations Support Team) who searches social media for storm reports. The NWS Austin/San Antonio office is very active in this area. The even encourage folks to use the hashtag #eWXSpotter. Social media reports allow you to send pics, which the NWS offices love, because the can see what someone is reporting. Twitter is better for reports than Facebook as Facebook forces an algorithm timeline instead of letting you see things chronologically. Every time in 2019 that I made a report via Twitter I got an almost immediate like or response from the NWS WFO account. I will note that this method of reporting is highly dependent on WFO office staffing at the given time.