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/stable/2324098?seq=1#page_scan_tab_contents

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).