The only approved inhaled vaccine is the flu vaccine delivered intra-nasally. It uses a live attenuated virus. There are a whole list of people who should not receive it because it's a live virus, and it works better for children, but only is 40% effective for adults.
The main issue is
The intranasal LAIV, recommended for children above the age of 2 years, induces a broader immune response wherein protection is not antibody mediated and probably involves undefined multiple correlates of protection.
and that's with a live virus.
How well would a dead inhaled virus work? Well, we have some data from MERS that suggests that vaccination with whole dead virus can increase lung pathology when exposed to live virus.
The implication of the current study is that application of an inactivated MERS-CoV vaccine for prevention of MERS in humans may carry a risk for lung immunopathology if subsequently exposed to MERS-CoV. The study also leads us to suggest that the extensive background of preclinical experience with inactivated SARS-CoV vaccines may be applicable to inactivated MERS-CoV vaccines.
So, your cheap vaccine may increase your risk of death or disease when exposed to the live virus.
Influenza Vaccination Strategies: Comparing Inactivated and Live Attenuated Influenza Vaccines
https://www.cdc.gov/flu/prevent/nasalspray.htm
Immunization with inactivated Middle East Respiratory Syndrome coronavirus vaccine leads to lung immunopathology on challenge with live virus
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5027702/
Are there known reasons why this would not work?
I think the more important question is - is there any evidence whatsoever that suggests this even remotely has a hope of doing anything beneficial at all. I'm curious about what would give you the idea that this could possibly be effective?