Yeah problem here is classification. If a specie reproduces exclusively by budding or similar processes, it's asexual by our definition (also by default it's a species more susceptible to diseases and unfavorable environmental factors). If the species has only one sex that produces both gametes, they're also not male, but monoic or hermaphroditic. By definition a species must have a female for there to be a male, because by default females have the most important gamete, males mostly contributing to genetic material. Thunk of it like a factory that simply needs one final piece to fully start. A great example of this is how female komodo Dragons can give birth to normal males (and in this case exclusively males) through parthenogenesis, being the largest animal known to engage in such a type of reproduction. It also means that, if they could magically ignore inbreeding-related problems and issues associated with a lack of genetic diversity, komodo Dragons could basically exist for as long as there was a single fertile female around to kick-start the species back from extinction.
Also worthy to note that in terms of how we do things in biology and determining sex, external appearance is borderline meaningless. In hyenas for example the females have a pseudo penis and many traits usually associated with male animals, but are nonetheless females. Same can be said for human males with the Y chromosome and a female appearance.
So summing up: can a fully monoic exist? Yes, and they already do in our world, as do species which are capable of various degrees of both sexual and assexual reproduction and can favor one over the other depending on the situation they're in (by various degrees I mean whether they can or cannot rely exclusively on reproduction without a partner to survive in an ideal scenario where low genetic diversity aren't noteworthy). Can a male-only species exist though? Nope. They can't, and not because of biological limitations, but rather simply because of how we define this stuff.
If you still want them to be called males, the best solution is to change how your own world defines a male organism when compared to our own. Alternatively, if they were a species discovered by another species that does have 2 sexes, you can pull out the hyena card and have a species entirely composed of females that naturally evolved to have traits "more associated with males" (and I put this between commas because that's something more associated with us mammals but not universal. In many bird species, especially raptors such as harpy eagles, the female is usually the sex that's known to be bigger and more physically powerful rather than the male). This results in the species being mistaken for males despite being originally the female of the species, the males now long extinct as the species adapted reproduce exclusively through parthenogenesis (much like its rumored that hyenas were originally thought to be a monoic species, but that may be just a myth).