The phrase reach out and touch is is an idiom because it deliberately conflates two or more different meanings of the word touch and cannot be understood based on the literal meaing of the words used.
The word touch in the phrase is a verb form, and is preceeded by the phrase reach out. In this context, it suggests the following meaning (transitive verb, meaning 1)
To cause or permit a part of the body, especially the hand or fingers, to come in contact with so as to feel: reached out and touched the smooth stone.
However, the context in which the phrase reach out and touch is used rarely, if ever suggests an actual touching. Rather, in the tagline that made it famous, as described in other answers, the contact was at long distance.
Instead it could more readily convey being in touch. This phrase uses touch as a noun form. The phrase in touch means (noun, meaning 14)
The state of being in contact or communication: kept in touch with several classmates; out of touch with current trends
Finally, the word touch can convey an emotional, rather than physical contact (transitive verb, meaning 10).
To affect the emotions of; move to tender response: an appeal that touched us deeply.
Given the layers of meaning, and the non-literal usage, it is an idiom,