I wonder if there is a closed-form, or clean upper bound of this quantity: $\mathbb{E}[|X/n-p|]$, where $X\sim B(n,p)$.



A clean upper bound for the expectation of a function of a binomial random variable
This is the mean absolute deviation (MAD) for a binomial distribution, divided by $n$. The expectation is hence $$2 \, (1-p)^{n+1-\lceil np \rceil} \, p^{\lceil np \rceil} \, \binom{n-1}{\lceil np \rceil-1} \;.$$
See [this paper](https://www.cs.bgu.ac.il/~karyeh/sharp-mad.pdf) (Berend & Kontorovich 2013, doi: 10.1016/j.spl.2013.01.023) for bounds and a reference for the above expression.


