Ohio certainly is restrictive, but not neccesarily prohibitive.
Section 4511.01 of the Ohio revised code defines a motorized bicycle as any vehicle with two tandem wheels or one front and two rear wheels which is capable of being pedaled and is also equiped with a helper motor. The law specifies a helper must:
Have a piston displacement of no more than 50cc.
Produce no more than one-brake horsepower.
Propel the vehicle to a maximum speed of no greater than 20 mph on a level surface.
Yes, the state does have a list of approved manufacturers', but they are not limited only to those companies.
Section 4501-23-19 Assembly by person other than a manufacturer.
4501-23-19 Assembly by person other than manufacturer.
Nothing in these rules and regulations shall prohibit a person other than a manufacturer from constructing, assembling, or equipping a vehicle so as to conform to the specifications of a motorized bicycle. Such person shall, however, comply with all of the rules contained in Chapter 4501-23 of the Administrative Code to the same extent as is necessary of a manufacturer.
R.C. 119.032 review dates: 09/12/2006 and 03/09/2011
Promulgated Under: 119.03
Statutory Authority: R.C. 4511.521
Rule Amplifies: R.C. 4511.521
Prior Effective Dates: 4/1/78
If you fill out the following linked form, for about $30.00 you should be able to produce your own bike, providing it meets the states safety requirements.
http://publicsafety.ohio.gov/links/bmv4312.pdf
Some states allow your Soc. Security # to be used as a tax number, that you'd have to check further.