I think the question implies a preference for sources of real time vector data, but given the title, I thought it a good place to remark on any and all sources of real time GIS data.
USGS Real Time Streamflow Data
The US Geological Survey real time streamflow data provides a fun source of information. Clearly the gauges are points corresponding to changing streamflow values. From the USGS site:
Current data typically are recorded at 15- to 60-minute intervals,
stored onsite, and then transmitted to USGS offices every 1 to 4
hours, depending on the data relay technique used. Recording and
transmission times may be more frequent during critical events. Data
from current sites are relayed to USGS offices via satellite,
telephone, and/or radio telemetry and are available for viewing within
minutes of arrival.
Weather data applies, too. I use a WMS call like the following in conjunction with a Google Map to quickly get local weather radar images without ads. Here's an example of the URL. I don't know how often the feed refreshes, officially, but rapid refreshing during a big weather event reveals it to be very continuous and real time. There is no telling what you'll see if you click this (if we have clear skies in Springfield you'll just get a transparent PNG):
You can investigate the various NOAA services here. It's a cool site with several free services and some descent examples.
NWS Data (a division of NOAA, but tasked with warning and advisory responsibilities)
The National Weather Service also provides some free services. I backed into the URL a little and got this page with a raw list of WMS layers you can grab. I followed the link to one of them, opened Fiddler, then selected View In.. ArcGIS.com Map, and I was able to trap a WMS URL similar to the one above and monkey with it until I got something I wanted. This example takes a Lon/Lat Bounding Box (EPSG:4326) and returns a PNG map image projected to North Aamerica Lambert Conformal Conic (EPSG:102009). This approach is repeatable for any layer you want, such as the warning/advisory polygon areas, but the radar is more likely to always show something so I used their radar: