I have missing data on my covariates of interest in my samples (a human longitudinal study) on which I am going to run limma.

Multiple imputations in R give multiple datasets. Is it advisable to run limma using each of these imputed datasets and then average the estimates somehow that I get from limma or is there a better way of doing this that I am missing out on?

The missingness is between 10 to 13% in these covariates.

limma is not designed to be used with multiply imputed covariates, so I can't offer a solution that uses the multiple imputes.

Averaging the results from multiple runs defeats the purpose of multiple imputation. If averaging was your plan, then it would be better to use mean imputation in the first place instead of multiple imputation. With multiple imputation, you could run limma on a few different imputations and see if the results remain stable. If they do, then you can use any of the results.