I might be running the risk of offending people with disabilities here - if that happens, it's entirely unintentional and I apologize in advance.
I don't think they'd have problems remaining at the top of the food chain, but what I'm sure of is that they'd be much more cooperative with each other. This isn't about whether some creature evolved through natural selection would survive - it's about removing a very important adaptation from an existing one abruptly.
I assume their culture would change to make room for the very high need they'd have to help each other - they wouldn't be helpless if alone, especially with the proper technology (such as wheelchairs), but the same dangers would escalate a lot faster (an example is running away from dangers, as you mention).
Rowanas however makes an excellent point about running, in the comments. Since arms do extend beyond the hips, running can be performed with hands alone, albeit with a smaller stride - check his comment for the full idea.
This doesn't necessarily mean they would be less likely to be hostile or unhelpful, just that it would be much more frowned upon between them.
Overall, if they are already aware and capable of tool use and technology, I don't think it'd be long until they've adapted their tools and life to work with their lack of legs. Transportation wouldn't be that huge of a problem if everything is made to work that way.
Hunter-gatherers
If they're starting out at this level, basic problems need to be solved quickly, such as being in constant contact with the ground - rough terrain would be dangerous and hard to cross. Simple solutions such as chest-skis or chestplates made of wood could help. That of course requires cutting wood - it might be easier to do with a hoe-like tool instead of an axe, to take advantage of being on the ground for resistance and so it can be used without other aids.
They'd also probably smooth out the land around dwellings a lot more to make it easier to move around. Housing would probably be quite a bit shorter too - it would be easier to build and use.
Foraging shouldn't be much different, but picking food off trees would require a lot more effort than it does for someone with legs, since there's nothing to hold you as you grab for a higher place - it can be worked around using leather harnesses however.
Which brings us to hunting - basic animal food would probably caught with traps, with larger animals being herded into traps or hunted in packs to prevent dangers from getting attacked by wild boars while lying down or getting trampled. Alternatively, herds that wander close to trees could be killed from above by climbers.
Addendum by tepples
This contrasts with the solutions that real-world humans of this era found, one of which was stalking mammoth. Having discovered that, inferior solutions or those harder to come up with would have been overlooked or just not used. While humans without legs would not be able to run behind mammoths throwing spears with their arms effectively, they can still stalk prey covertly in groups, trap it or use other techniques that for us are inefficient.
This would limit their food supply, assuming their approaches are inferior, which would limit the sizes of their populations and hence the growth rate of their species. But I don't think that it can be said to be definitively impossible to survive the hunter & gatherer phase, or to say that not having legs is a death sentence for a species of otherwise human capability.
Rowanas has commented on the tool usage and climbing capabilities as well, check the comments section.
Farming
Plowing can be performed through harnesses, but the taller plants might introduce difficulties. Perhaps frames, like scaffolding, could be built within crops to allow for this, like shelves along the vegetation. This, like many of the previous approaches, would probably make performing these activities hard enough to delay progress, unless said humans were aware of technology like this and could come up with these ideas quickly.
Further progress
I think that once there's enough wood to create wheelchairs, wheelbarrows and the like, these humans would progress a lot faster. They might have a problem finding animals small enough to tame for use as beasts of burden (transportation and field work).
Given the need, people can learn to walk on prosthetics and crutches fairly well - the only problem here would be that they'd never have learned how. Unless they change genetically however, they should maintain the balance adaptations we have and all the rest so they could learn to walk on prosthetics and crutches quickly enough to use them. This wouldn't allow them to run, but with entire populations of people using these things daily, the techniques on making them comfortable and robust would develop quickly.
Given time enough to develop metallurgy, they should be able to create more complex spring-loaded mechanisms to allow for stride-capable prosthetics. This wouldn't get around to them having to use their hands to walk (since they have no legs at all) but it would make it easier and some might manage to run (although at this point it wouldn't be much help).
However, walking like this would probably be just for outside work - by this point, everything the use should be accessible without having to walk - perhaps they'd have wheelchairs at most for every-day use, which are easier to use and more practical.
Modern levels
I don't think much would stand in their way after they're comfortable moving around and dealing with the natural surroundings. Their buildings and technology would adapt and in order to use your arms properly you don't need to stand on legs, just be upright so you've got free room to move. I wouldn't put industry and even space-grade technology beyond them. Legs are very important, but unless we're talking about primitive people, they should be able to cope just fine, but it would still take centuries at least, assuming they maintain some cultural memory of what the rest of human society was like.
