It sounds like you want the logit-normal distribution. This distribution shows up a lot in Compositional Data Analysis (CDA). CDA is often used in geology to measure the composition of minerals within soil or rock samples. The logit-normal takes a logit tranform of your random variable and this logit-transformed random variable is a normally distributed random variable. Formally,
where $X$ is logit-normal and $Y$ is normal. Multivariate extensions exist and are the more commonly used forms of the density.
If this is not what you want and you truly want a normal random variable that is restricted by a collection of constraints to always sum to 1 and have all entries be non-negative, you'll need to resort to other simulation techniques to get draws from the distribution. It is fairly complicated to perform these draws. John Geweke wrote a paper about doing this and Christian Robert also wrote a paper on sampling from this type of distribution.