Log-normality assumption in consumption based asset pricing
The typical two-period Lagrangian is

$$\Lambda = \beta^t\cdot \Big(\frac{c_t^{1-\gamma} -1}{1-\gamma} + \lambda_t\cdot \big[(d_t+p_t) \pi_{t-1} + \pi_{t-1}^0- c_t - \pi_t p_t - \pi_t^0 p_t^0\big]\Big) \\
+ \beta^{t+1}\cdot \Big(\frac{c_{t+1}^{1-\gamma} -1}{1-\gamma} + \lambda_{t+1}\cdot \big[(d_{t+1}+p_{t+1}) \pi_{t} + \pi_{t}^0- c_{t+1} - \pi_{t+1} p_{t+1} - \pi_{t+1}^0 p_{t+1}^0\big]\Big)$$ 

The first order conditions with respect to $c_t, \pi_t$ are

$$c_t^{-\gamma} = \lambda_t \implies ... \gamma\ln \frac {c_{t+1}}{c_t} = \ln \frac {\lambda_{t}}{\lambda_{t+1}} \tag{1}$$

$$-\beta^t\lambda_tp_t + \beta^{t+1}\lambda_{t+1}(d_{t+1}+p_{t+1})=0 \implies  \frac {\lambda_{t}}{\lambda_{t+1}} = \beta \frac{p_{t+1} + d_{t+1}}{p_t} \tag{2}$$

and so, using also the definition of the gross return,

$$\ln \frac {\lambda_{t}}{\lambda_{t+1}} = \ln \beta + \ln R_{t+1} \tag{3}$$

Combining $(1)$ and $(3)$ we get

$$\ln \frac {c_{t+1}}{c_t} = \frac 1 {\gamma}\ln \beta + \frac 1 {\gamma}\ln R_{t+1} \tag{4}$$

So we see that at the optimal path, consumption growth is a direct affine function of the log-risk returns. This among other things implies that their correlation coefficient is equal to unity.

The normal distribution is closed under affine transformations (alternatively, under scaling and shifting), so if we assume that log-risky returns are normally distributed, then consumption growth is also normally distributed (with different mean and variance of course).

Note that although in general, the joint normality assumption is an additional one to be made when two normal random variables are not-independent, here, the fact that the one is an affine function of the other guarantees joint normality. By Cramer's condition for bivariate normality, it must be the case that all linear combinations of two normal random variables have a univariate normal distribution. In our case we have (generic notation) the random vavriable $Y$ and the random variable $X = a+bY$. Consider

$$\delta_1X + \delta_2 Y = \delta_1(a+bY) + \delta_2 Y = \delta_1a + (\delta_1b+\delta_2)Y$$

So for any $(\delta_1, \delta_2)$ (except the zero vector which is excluded a priori), $\delta_1X + \delta_2 Y$ follows a normal distribution if $Y$ does. So it is sufficient to assume that log-risk returns follow a normal distribution to obtain joint normality also.