I would like to test whether an intervention boosts self-efficacy at time 1, which then lasts through self-efficacy at time 2, and at time 3, etc, to then have an impact on medication adherence at time 5. Is serial mediation an appropriate method for this question? I would be using the PROCESS macro.
First, yes it can. You will be in a slightly different framework though, which is longitudinal mediation. It would be consider a cross lagged mediation panel, but a weak one, given the design presented (a single measure of adherence?).
My two cents on your analysis : if your design enable it and are willing to use structural equation modelling, I would consider a latent growth model of self-efficacy, from which I would use the latent variable as mediators. It seems much better and appropriate to answer your question.