It seems that you are strugling with an adequate assumption about the distribution of the response variable. Classical linear regression and classical ARMA-models assume that the response variable, has support on all the real numbers $(-\infty, \infty)$. Often the response is also assumed to be normally distributed. This is clearly not the case in your application.
I would first try to disregard the (potential) time interdependence of the data and fit a Beta-regression. The Beta-regression is a Generalized Linear Model (GLM) assuming the response variable follows a Beta-distribution, when conditioning on co-variates. The Beta-distribution is a very flexible continuous distribution on the unit interval, $(0,1)$. This answer has some good references: Regression for an outcome (ratio or fraction) between 0 and 1.
If you find that there is significant serial correlation in your response variable that the co-variates cannot account for, I would look into Beta-ARMA models of Rocha & Cribari-Neto (2009) or Guolo and Cristiano Varin (2014). Guolo and Cristiano Varin (2014) is probabely the easiest one to get started with since they have a nice example in R where they fit a Beta-ARMA model to illness percentage over time.