Here are a few aspects that have not yet been mentioned by
Felipe, Joe and Vesselin.

a) Assuming GRH, 2 is a primitive root mod $p$ for a positive
density of primes. (The density is 0.373955, known as Artin's constant).
For a great survey on this see Pieter Moree,
Integers, Volume 12A (2012), A13: Artin's Primitive Root Conjecture -- A
Survey --

Moreover I wouldn't be surprised, if for each fixed $L$ there is a
positive density of primes, where the order of 2 mod p, $\text{ord}_p(2)$ is $\frac{p-1}{L}$.

Assuming GRH, the average value of the order is quite
large, see for example Thm. 2 of Kurlberg and Pomerance:
ON A PROBLEM OF ARNOLD: THE AVERAGE
MULTIPLICATIVE ORDER OF A GIVEN INTEGER

https://math.dartmouth.edu/~carlp/arnoldfinal.pdf

b) Your question whether $\frac{p-1}{\text{ord}_p(2)}$ can actually be
bounded has appeared in the Monthly a while ago:

E3216
Jon Froemke, Jerrold W. Grossman, O. P. Lossers
The American Mathematical Monthly, Vol. 96, No. 4 (Apr., 1989), p. 361
http://www.jstor.org/openurl?issn=0002-9890&volume=96&issue=4&spage=361&date=19$

In short, the published solution gives an elementary proof based on
Dirichlet's theorem on primes in progression, and mentions that other
proofs exist that are "not completely elementary",
(e.g. including Chebatorev).