/u/conquete_du_pain explains how Hannibal (according to Cornelius Nepos) carried poison with him to keep himself from being captured alive. While I'll go ahead and link to my own comment on that subthread to mention a few of the problems with taking ancient biography as literal fact, it would hardly have been impossible. Poison was a well-known means of suicide in classical antiquity, particularly among the powerful; according to Livy, poison is a thing "which kings usually have in reserve against the vicissitudes of Fortune". The risk of torture and degrading treatment in captivity aside, dying on one's own terms, rather than living on another's, was a common theme in Greek and Roman philosophy. Seneca, for instance, applauds suicide in such situations. It was often considered courageous; the poet Horace, for example, praises Cleopatra for having the courage to take her own life after capture (though his motives might not have been so direct).

(Nota bene: in both of the specific cases OP asks about, Hannibal and Boudicca, we're getting the story secondhand through Roman sources. I don't have any information on philosophical attitudes towards suicide among either the Punics or the Iceni specifically. More, we're reading their stories through a Roman cultural filter that sees suicide to avoid capture as the appropriate end for a courageous enemy, and there are alternate death stories available for both - Pausanias says that Hannibal died from an infected wound, rather than poison, and Dio says the same of Boudicca. Take the stories with appropriate levels of salt.)

As to where poison might be obtained for suicide: most poisons used in the classical world were plant-based (hemlock, wolfsbane, etc.) and easily found growing wild. Compounded poisons could be readily purchased from doctors or herbalists; poisoned spears and arrows were used for hunting, and many poisons, in small doses, were considered medicinal (for instance, opium was used as a painkiller, but people suffering from incurable diseases were known to commit suicide via opium overdose). Anyone who wished to carry poison, just in case - and it would not be looked down on for a powerful person to do so - or kill themselves with poison would have no trouble obtaining a dose.

Finally, I'll note the cautionary tale of Mithridates VI of Pontus, who, as the story goes, dosed himself with small doses of poisons and antidotes all his life in order to protect himself from poisoning. When he was overthrown, he attempted to poison himself, but failed, because his resistance to poison was too great. He died by the sword, either in the hands of a friend or of his enemies, a less dignified fate than the one he had planned for himself.