I think this question can only be answered if you know not only the constraints on species counterpoint (as the OP clarified to Westergaard's formulation), but also the constraints on what constitutes a "proper" cantus firmus.
Westergaard (I'm assuming we're talking about his tonal theory book) is working with a set of rules of species counterpoint that are designed to introduce tonal theory. Thus, there are a lot of unwritten assumed constraints on his first-species counterpoint that are basically laying the groundwork for coherent harmonic progressions (as Fux does as well). If you literally read only the rules he officially states for first species, and you assume a cantus firmus that doesn't wildly violate basic tenets for melody or bass line he has elsewhere laid out, I don't see how you could possibly fail to have a possible counterpoint line. In some cases, it may not be very musical and might not incorporate the tonal characteristics he seems to be aiming for, but in terms of strict adherence to his species constraints, I don't see how you could generate a cantus firmus that follows his melodic principles and wouldn't have a counterpoint solution. (Westergaard's constraints are a lot more lax than many authors.)
For an example of what I mean by "unmusical," one could simply imagine a cantus firmus ending with a stereotypical octave leap down, then fourth up for a cadence, but then insisting on creating a counterpoint below that CF. The result would sound absurd, but I don't think it would technically violate Westergaard's explicit constraints on first species. It would certainly violate the spirit and intent behind his method, though.
Introducing further constraints would lessen the chances of producing something unmusical but still "acceptable" to the explicit rules, but if you did so, then the reasonable constraints on a "valid" cantus firmus should probably be tightened up too, resulting in little chance of a CF with no solution.
This latter idea has been thoroughly investigated by computer programs modeling solutions since the 1950s. (A brief summary of that history, with some useful sources at the end, can be found here.) One of the most useful articles that I recall reading on this topic is David Lewin's "An Interesting Global Rule for Species Counterpoint" (In Theory Only 6/8, March 1983). Lewin's "global rule" produces more generally "musical" solutions for a given CF, but it does place heavy constraints on solutions. In general there, one needs to work backwards from the final cadence, which is the most constrained and trickiest part. Again, I think Lewin mostly presupposes a "reasonable" cantus firmus that doesn't violate general late Renaissance melodic principles. But, for example, if you tried to combine Lewin's rules with some of Westergaard's more "tonal" cantus firmi, and then insisted on doing something silly like using Westergaard's bass-line-like cantus firmus as the top melody, then of course you couldn't find a viable solution. But in that case, Westergaard's CF would violate Lewin's melodic principles to begin with, so I'm not sure that's a valid test.
So, in general, I'd say even with severely constrained species rules, as long as the cantus firmus is "reasonable" (and following the same severe constraints in its own generation), counterpoint solutions are probably available. However, to speak of mathematical proof would require a more formal statement of the rules. While the rules for generating counterpoint are usually fairly explicit in many species methods, the rules for creating a "valid" cantus firmus are often not stated as explicitly, making this question impossible to answer for many cases.
