I've never been there, but here's what I found on the net for ya:
Here's a sporting goods store in Bristol, VA, perhaps you could call them and ask about fishing there
http://www.mountainsportsltd.com/ Clear Creek Golf Club
http://www.bristolva.org/ClearCreekMainPage.htm has a 35 acre lake and fishing pier
http://www.bristoltn.org/steelecreekpark.cfm this place has a nice lake, but doesn't say anything about fishing, but my be worth a call to find out
Also, a creek, known as Beaver Creek, flows through town. It may have trout or smallmouth bass perhaps. There are other creeks around too, so they probably have smallmouth bass.
Other potential spots nearby:
Beaver Creek Impoundment, 5.2 miles from Bristol, VA - has rock bass, walleye, and crappie. I'd bet it has smallmouth and largemouth bass too.
Clear Creek Lake, 6 miles from Bristol, VA - has brown trout, bluegill, and muskie, probably has bass too.
South Holston Reservoir, looks like it's about 10 miles away, but has all kinds of fish in there, bass, muskie, walleyes, trout, etc.