There are a few ways you could go about this:
1. Call property management companies
In my opinion, this is probably the best way to do it. PM companies in your area should have rental data already, as they'll be tracking it every month. You can ask them about rent trends in the market as well as how they anticipate rents will be affected by current events.
2. Use online resources
This may be what you're already doing. You can get rent information from websites like Apartments.com, Rent.com, and zumper.com. I don't think those sites have an easy way to see historical data, but if you track it over time then you'll be able to see the long-term trends.
3. Conduct your own market survey
This is probably the most time-consuming option. It involves calling individual properties, property managers, or owners, and asking about their current rents. You can also ask how rents have changed over the past few months, but they may not give you that information. You'll have to repeat this over several months to start to see meaningful trends, but of course things may change dramatically over the next few months.