Short answer: Yes, but I don't think you'd want to.
Nukes would be cheaper so it's feasible, but nukes can be significantly more dangerous.
Wikipedia actually has a calculation for something along these lines: (Paraphrasing)
With the "Momentum Limited" Orion, a 50,000,000 kg payload could be
brought from Earth to Alpha Centauri and would cost around \$367
Billion USD. That's about $7,340/kg.
Source: Wikipedia: Project Orion
The SpaceX Falcon 9 rocket can put 4,850 kg into Geosynchronous Transfer Orbit (it's a cakewalk after getting to GTO) for \$61.2 million. That's about $12,618/kg. Add an additional chunk to get us further into space and we're even more expensive (I can't give an exact figure).
The real problem is that nukes give off loads of radiation. And you can't really get much away from it. You could shield the effect with lead, but the more lead, the more it costs, and the less you could bring up with you. Traditional rockets don't have this issue.
Additionally, traditional rockets have to be made small just due to the cost of leaving Earth. The Orion craft mentioned above from Wikipedia, you'd be putting a lot of your eggs in a single basket. Specially 10,309 eggs for each traditional rocket into a single orion basket.
Most rockets have a 98%+ success rate. For the same weight in the orion ship, you'd have around 10,309 traditional ships. Around 206 would fail (around 1,000,000 kg). At an average of ~80.7kg/person, at most that's 12,391 lives lost (likely less due to luggage)
If the orion craft breaks, you'd lose all 50,000,000 kg. At most, that's 619,578 lives lost. That's quite a lot of lives.