Services like Google Scholar can provide you with the h-index of a given author at present. Is there a way or a service that provides a time series of the h-index (or the number of publications) as a function of time, or even provides a plot of this time series?
Scopus has an author search option. You can then examine a range of graphs that show academic output over time. Many universities have a subscription to Scopus.
As with any citation and publication based metrics, the values are contingent on the citations and publications included in the underlying database. Scopus is fairly broad, but there is quite a lot of quality control. So the numbers will be quite a bit smaller than for example Google Scholar. Another option might be PublishOrPerish which is a downloadable free program which uses Google Scholar as the underlying data source.
Below I show an example for Albert Bandura (one of the most highly cited psychology researchers) using Scopus.
Citations per year
Documents by year
h-index by year
I could not see any easy way to just get the h-index by year. However, the h-index tab, does allow you to filter on different years. So you could manually, change the end year and repeatedly obtain the h-index to see how it went up each year.
I am not actively aware of any tool that does this, although I vaguely remember seeing such timelines for various bibliometrics tools. However, if you are able and willing to do a little bit of coding, it should be reasonably easy to code this up yourself.
For instance, SCOPUS has an API that you can use to get all citation data for an author. From there, you should be able to can just browse the publications that cited this year to year backwards, remove all publications whose publication date is after the date you are currently looking at, and recompute the h-index. A Python script to this end should be quite easy to set up.