Training without using the inhaler will just mean that you won't get as much oxygen as you need to exercise with maximum effect. Nowadays I use my inhaler (Ventolin) 20-30 minutes before I go running and then I am usually fine (even when I am running for a couple of hours); in my case the asthma starts to affect me within about 15 minutes if I don't use the inhaler before exercise. Note that using the inhaler BEFORE exercise is usually a good idea (although I guess you know this), if I use the inhaler while exercising my heart rate goes up slightly (side effect of the Ventolin) and I have to reduce the pace for a while to get back into the aerobic zone.
Anyway, as far as I know there is NO evidence that training without medicine will reduce the effects of asthma. Increased fitness certainly helps reduce the symptoms
but AFAIK the severity of the asthma is basically governed by genetic factors; for many (including me) the situation improves with age and in some cases it even disappears (usually in the teens) but I don't think there is anything you can really do to treat the underlying cause (which isn't fully understood).
Also, unfortunately I think it is unlikely that you will be allowed to join special forces, I wasn't even allowed to do my military service which was mandatory in Sweden back then; and that was just basic training. I suspect US special forces have even more stringent requirements. Although I should point out that my asthma can also be triggered by other things (horses, mould etc), maybe things would have been different if it was only exercise induces (although I doubt it).