Can research papers postulate a theory without proof or assurance of correctness? What I mean is, can research papers be written on a problem for which there is a solution postulated but without any actual proof or working model to show that it does work?

To give context, the problem is in the context of Computer Software, and the solution provided is not backed up with a proof of concept.
Sure. All kinds of papers get written and submitted all the time. The best ones have strong empirical data or formal proofs to back them up. Others ... are based pretty much on opinions or hot air.

Of course, you are likely actually asking whether such a paper has a chance of acceptance at a reasonable venue. Here, the answer depends on the field as well as on the type of conference or track you are submitting to. In fields such as math, a paper where you are basically just speculating will (based on my limited knowledge of mathematical research) have close to zero chance of acceptance in pretty much all journals. Computer science is often more forgiving, particularly if the idea is interesting enough and the reviewers believe that it will work. Further, in Computer Science there are (even well-respected) conferences and tracks that specifically target position papers, essays, or "reflection" papers (see for instance Onward!, co-located with OOPSLA). For Onward!, no real scientific evaluation is required, however, you will need something to substantiate your ideas (e.g., some proof-of-concept implementations, initial studies, ...).

One non-obvious fact about such "opinion-based" papers is that they are actually much harder to write than ones that are based on hard science. If you conducted a thorough study on a subject, your data will go great lengths for you. Even if you don't tell the story of your research in the best way, if you have good, hard data, your results alone may carry you. If you write an opinion piece for a competitive conference such as Onward!, your writing needs to carry you all the way.