Probably "normal model" means a process in which every (finite) linear combination of $(Y_t)_{t\,\in\, \text{index set}}$ with constant (i.e. non-random) coefficients has a Gaussian distribution. The main point of the exercise is probably intended to be to prove that $e_t$ are independent.
Suppose $Y_1\sim\operatorname N(0,1)$ and $Y_2 = X Y_1$ where $X=\pm1$ each with probability $1/2.$ Then $Y_1,Y_2$ are each normally distributed and uncorrelated by NOT independent and $Y_1+Y_2$ is not normally distributed. And let $Y_3= \begin{cases} Y_1 & \text{if } |Y|>c, \\ Y_2 & \text{if } |Y|<c. \end{cases} \qquad$ Then for a certain value of $c>0,$ $Y_1,Y_3$ are normally distributed and uncorrelated but NOT independent and $Y_1+Y_3$ is not normally distributed. The point of these examples is that separately normal does not imply jointly normal, i.e. does not imply that this is a normal process. "Normal model" is the same as "jointly normal", i.e. every constant linear combination is normally distributed. When random variables are jointly normally distributed, then uncorrelatedness is as strong as mutual (not just pairwise) independence.