Ten crews were sent to Greenway Polymers – formerly known as Johnson Aggregates & Recycling Ltd, Greenway Waste Recycling and Pink Skips – on Rock Road, which sits alongside the motorway.
Fire officers said the blaze involved plastic bales which were alight.
Shropshire Fire and Rescue Service urged nearby residents to keep doors and windows shut.
Police closed Waterloo Road and Overdale Road and said they will be shut "for some time", with drivers advised to avoid the area.
Clouds of thick black smoke could be seen for miles, with fumes affecting residents in Ketley and Wellington.
Shropshire Fire & Rescue Service said there would be traffic disruption and urged motorists to take an alternative route.
Nearby residents Jon Borl and Dannie Tudor say they noticed the smoke about 12.15pm. They said they have previously had issues with the recycling centre in terms of heavy goods traffic.
Firefighters drew up water from pipes underneath both ends of Waterloo Road, with hoses snaking all over the road and through fences adjoining the site.
Crews later moved onto the hard shoulder of the M54 as the fought to stop the blaze from spreading.
They urged motorists to reduce speed and give plenty of room to crews as they passed the scene, with lane closures in force.
Several pedestrians gathered on the side of Waterloo Road, directly opposite the entrance to the site, to watch dozens of firefighters tackling the blaze.
One woman said she was told by an officer that they could be there for the next few days. They were eventually moved on and the road was closed completely with tape and barriers.
Some residents nearby watched the smoke pouring from the warehouse from their gardens.
A West Midlands Ambulance Service spokesman said: "We were called to reports of fire on Rock Road at 12.16pm, one paramedic officer attended the scene. However, there were no patients that required treatment and we were stood down at the scene."
INCIDENT 📟 Crew now committed to firefighting duties at large fire in #Wellington 🔥🚒🚒 Crews working on #M54 to prevent fire spread 👩🏻🚒👨🏻🚒💦 Lots of smoke from the fire 🔥 Please reduce speed and give plenty of room to crews as you pass the scene 🙏🏻 Lane closures are in force 🚧 pic.twitter.com/1MzXxTApeg— Market Drayton (@SFRS_MDrayton) April 26, 2021
INCIDENT 📟 In attendance along with numerous other @shropsfire crews at large fire at commercial property in #RockRoad #Wellington 🔥🚒🚒🚒🚒🚒🚒🚒 Large column of smoke being produced, please avoid the area if you can, road closures in operation ❌ @TelfordCops @WestMerciaOCC pic.twitter.com/S2eBJfsd41— Market Drayton (@SFRS_MDrayton) April 26, 2021
Additional reporting by Rob Smith