Applying fitted mixed model to new data but without random effect data

I have developed a linear mixed model using the lime package and three fixed effect predictors and one random effect. In the absence of better information, the random effect was set simply as the name of the various groups studied. The model performs very well. Now I would like to use these effect estimates to derive a generalizable model that could be used by anyone to make predictions of our response variable of interest for groups for which users will be able to quickly measure their fixed effect variables. Problem is that, to propose such generalizable model and for it work, users will also need to be able to estimate the intercept of the random effects for their groups. Looking at the intercept of the random effects and the groups they represent, I cannot see that they follow a pattern I could use to inform this process. Is there a way to estimate the the intercept of a random effect for such groups?

Many thanks for any thoughts!