What is the halacha regarding the fast if a firstborn is sick on erev Pesach and can't attend a siyum?
A sick person might be exempt from the fast.
R Mansour at dailyhalacha writes
Firstborns who are ill or otherwise frail are not required to fast, even if they do not attend a “Se’udat Misva.”
This fast is less strict than other fasts. All of the other fasts were instituted by the Sages, but Ta’anit Bekhorot, a custom adopted by many bekhorim, was never instituted as a binding obligation by the Sages. It is therefore customary to be lenient. For example, if one suffers from a headache, or from a pain in his eye, he is exempt from Ta’anit Bekhorot, even though he is not considered sick and would not be exempt from other fasts.
Alternatively, following R Ovadia Yosef's opinion that completing a tractate of mishna with commentary is enough for a siyum on Erev Pesach (see 5th comment here or this for more), the sick person could also learn a small tractate of mishna and make a siyum alone, or to get a friend come and do this for the him. Of the smaller masechtot with relevance to Pesach are e.g., Moed Katan with 24 mishnayot or Hagiga with 23.
As everything you read here, ask your local rav for a practical ruling if you want to apply it in practice.