Is the sample proportion ($\hat p$) a random variable? Since it should vary from sample to sample, I suppose it should be a random variable. But if it is, when we write the variance of sample proportions, should we write an uppercase P-hat as the index notation of sigma?
But in most textbooks and formula tables, it seems that a lowercase p-hat is usually used



A statistic is a function of a random sample, therefore it is also random variable.
Like Tanner Phillips says in his answer, the frequentist school of statistics establishes a difference between a population and a sample taken from the population. Population parameters are [always] constants and to estimate those parameters we sample from the population and compute a statistic or estimate. Unless the entire population is sampled, the statistic is not a constant, it's a variable as random as the sample.
In the case of proportions, the population proportion is usually represented by a lower-case $p$ and the sample proportion by the very same symbol plus a marker. The sample proportion is many times noted as $\hat p$ but the hat-p is not a standard notation for proportion estimator, it's a frequent one, not more.
I can not find examples right now but I think that the convention of noting the sample proportion as $\hat p$ is not universal and is even a relatively recent one. I have also seen $p^{\star}$ and $\widetilde p$. And sometimes, not frequently, when the population proportion is noted with an upper-case $P$ so is the sample proportion.
