Are there any (chemical) processes occuring inside a seed while it's in a dormant state?
We can consider two processes that a seed might care about:

(1) Ordinary chemical reactions that just happen in the cytoplasm of the seed cells. It's very hard to stop chemistry! Even in the freezer, chemical reactions occur (slowly). This is why long-term storage of biological specimens is generally in an ultra-cold freezer (-80C). Chemical reactions could degrade DNA and proteins and prevent seed germination, so seeds take steps such as drying to reduce the chemistry going on. But, clearly this just slows down the reactions, as seeds of all types gradually lose their ability to germinate (some faster than others). Spore-forming bacteria are the most successful organisms in this regard: they drastically dehydrate the interior of the spore and surround it with a highly protective coat, and there is evidence that some can literally remain viable for millions of years.

(2) Seed metabolism: enzymatic reactions and the normal biochemistry of the cell. "Dormancy" means in part that seed cells shut down processes like transcription and translation and minimize their metabolism. But, research suggests that the metabolic activity of seeds is not zero - among other things, there are active enzymes that are involved in repairing damage to DNA and proteins.