The following RV Parks are on the east side of Austin with easy access on the 130 Toll Road and easy access to downtown via the 130 Toll Road:
1. Near the intersection of Hwy. 71 and Hwy. 183 and and about 15 min. east of downtown is McKinney Falls State Park:
TPWD: McKinney Falls State Park
2. Not far from the 130 Toll Road and not far from Hwy. 183 and gets nice reviews and about 15 min. east of downtown
Austin Oak Forest RV Park | Main
About 15 min. south of downtown:
Austin Lone Star RV Resort
Austin Lone Star RV Resort
About 30 min. west of downtown:
1. Emma Long Metropolitan Park (on Lake Austin)
http://austinparks.org/apfweb/park.php?parkId=247
2. La Hacienda RV Resort (on Lake Travis)
La Hacienda RV Resort & Cottages
Or about 30 min. south of downtown
Crestview RV
Crestview RV Campground | South Austin, Texas Campground | Full hookups, WIFI Internet and Cable TV
Or about 45 min. west of downtown
1. Pace Bend Park (on Lake Travis)
http://www.lcra.org/parks/developed_parks/pace_bend.html
2. Sunset RV Resort
http://sunsetrvresort.com/
About 30 min. north of downtown in Round Rock
Old Settlers Park =>
Old Settlers Assoc. RV Park Review and Rating
Or about 45 min. north of downtown in Georgetown:
1. Georgetown Lake Corp of Engineers - Cedar Breaks Park =>
U.S. Army Corps of Engineers - Georgetown Lake Home Page
2. Berry Springs RV Park =>
Berry Springs RV Park: Georgetown's Newest RV Park!
3. New Life RV Park =>
New Life RV Park Georgetown, TX
About 30 minutes east
Bastrop State Park (was one of our favorite places to camp). But I'm not sure of its condition since the wildfires went through the park.
TPWD: Bastrop State Park
Lake Bastrop (north shore)
North Shore Park
Lake Bastrop (south shore)
South Shore Park
Or about 45 min. south of downtown
Lockhart State Park
TPWD: Lockhart State Park
Hope this gives you some options for consideration.