The problem is that money would not be enough. Training people to the level they can inject it would still take incompressible time.
It's not impossible to train lay people to inject most vaccines that I've seen. Does the polio vaccine have to be injected into a specific organ or the bloodstream? Or is it subcutaneous or intramuscular? If the latter this is a solvable problem by organizing aid workers or by forming a small network of competent locals, distributing lyophilized vaccine, training, and having vaccination clinics.
The problem is that it takes a lot of community outreach and organization to have people actually show up for a schedule of 4 shots at different ages. https://www.cdc.gov/vaccines/vpd/polio/public/index.html gives the US schedule, four shots at, "2 months old, 4 months old, 6 through 18 months old, and 4 through 6 years old."
We do a LOT to make sure that people have taken those vaccines. Including checking that you have when you enroll in school.
Keeping something on ice is much easier than storing at a precise temperature range and not freezing it.
The oral polio vaccine (OPV)
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Should be frozen. If thawed but still cold--below 8 C (46 F)--refreeze immediately. Place in freezer on arrival. If thawed and warm--above 8 C (46 F) [1]
Maintain continuously in the frozen state--14 C (7 F) or lower. The vaccine may be refrozen. A maximum of 10 freeze-thaw cycles are permissible provided the total cumulative duration of thaw does not exceed 24 hours, and provided the temperature does not exceed 8 C (46 F) during the period of thaw. [1]
Vaccine in liquid state in unopened dispettes may be used for up to 30 days provided it has been stored at 2 -8 C (35 -46 F). A maximum of 10 freeze- thaw cycles are permissible provided the total cumulative duration of thaw does not exceed 24 hours and provided the temperature does not exceed 8 C (46 F) during the period of thaw. [1]
Cost of $0.11 - $0.20 per dose. [2]
Inactivated Poliovirus Vaccine (IPV)
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Should not be frozen. Refrigerate on arrival. [1]
Shelf Life after Reconstitution, Thawing, or Opening - Ampoule: Discard if not used immediately. Vial: Until outdated, if not contaminated. [1]
Cost of $1.50 - $3.79 per dose [3]
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[2] https://www.unicef.org/supply/documents/oral-polio-vaccine-o...
[3] https://www.unicef.org/supply/documents/inactivated-polio-va...