What's your engineering budget?
This is a hideously impractical undertaking. Most of the mass of an orbital rocket is fuel and the tanks to hold it; even though your payload is tiny, all the rest of that stuff is big. The smaller a rocket is, the harder it is to design it with the high fuel-mass-to-dry-mass ratio that is required to attain high speeds, because some elements (like electronics) don't scale with the size of the rocket.
Depending on what design assumptions you make, you will get wildly different results. Some arbitrary examples:
Scenario 1: two stage solid rocket, 90% propellant fraction.
Stage 1: 700kN thrust, 240s sea level Isp, 40 tons propellant. Contributes 3886 m/s ∆v.
Stage 2: 80kN thrust, 280s vacuum Isp, 5 tons propellant. Contributes 6581 m/s ∆v.
Scenario 2: single stage kerosene/LOX rocket using the SpaceX Kestrel engine, fuel tanks whittled by hand from carbon fiber by a team of elves on Adderall, total dry mass 100kg, of which the rocket motor is half. 96% propellant fraction, 2.4 tons of propellant, 31kN thrust, 317s vac Isp, total ~9600 m/s ∆v. (At the end of the burn for the Kestrel option, the rocket would be accelerating at better than 30g, so I hope those elves know what they're doing!) I don't think this is actually plausible, but it's theoretically possible.
And a real world example.
Scenario 3: Vanguard 1/TV-4 was a 3-stage rocket that put a satellite into orbit that weighed only about 3 soccer balls. First stage thrust 135kN. From my brief research, Vanguard and the Japanese 4-stage solid-rocket Lambda 4S seem to be the two smallest orbital launchers in history.
So depending on how you tweak the other variables, the needed thrust could be anywhere from 31kN to 700kN.
As for the ball, FIFA regulations say the air pressure in a soccer ball needs to be between 8.5 psi and 15.6 psi (above ambient), so presumably it can handle somewhat more pressure than that without exploding. Sea level ambient air pressure is about 15 psi, so if you deflated the ball before launch (to ambient air pressure), then took it into vacuum, it would then be at 15 psi overpressure - the high end of regulation inflation, so a very taut ball, but not exploded.