Typically, one comes across examples of exponential family distributions that have analytically computable normalizing constants. Consider the normal, beta, Poisson, etc. distributions.
However, I don't see why the normalizing constant would need to be tractable in general. Is there an example of an exponential family distribution with intractable normalizing constant?
here is an example that is nothing like the normal, beta, Poisson, etc. distributions
The sample space of the Ising model is based on a graph $G=(V,E)$ specified by a set of vertices $V$ and a set of undirected edges $E$. The sample space is $\{-1,1\}^V$. The density is $$\exp\left(\sum_{i\in V}Q^1_ix_i+\sum_{(i,j)\in E}Q^2_{ij}x_ix_j\right)$$
The natural parameters are $(Q^1, Q^2)\in \mathbb R^V \times \mathbb R^{V\times V}$ and the sufficient statistics are $(x, xx^T)\in \{-1,1\}^V\times \{-1,1\}^{V\times V}$.
The normalizing constant is the complicated function
$$\Lambda(Q^1, Q^2)=\sum _{x\in \{-1,1\}^V}\exp\left(\sum_{i\in V}Q^1_ix_i+\sum_{(i,j)\in E}Q^2_{ij}x_ix_j\right)$$
source: pg 3 of
