In my opinion, your premise is within realm of plausibility.
As wiki notes, Mithridatism is most likely to work against biological poisons that can be recognised by immune system resulting in immune system creating antibodies to bind and inactivate poison molecules before they can do harm.
You might have to drop "permanent" part of sedation for that to work. Usually anaesthetics and nerve-oriented poisons work by binding to receptors in nervous cells preventing propagation of signal, or preventing deactivation of once initiated signal. If unable to kill, eventually such chemicals become metabolised and once blocked pathways and receptors are cleared to function again (high but non-lethal concentrations can cause permanent damage in other ways). If poison is never metabolised then small incremental doses will keep building up since every molecule not intercepted by antibodies will disrupt one nerve cell forever, increasingly disrupting consciousness and bodily functions (think being intoxicated, not narcolepsy) until reaching lethal levels of disruption or rendering person/animal a vegetable.
One animal being able to ignore your plant's poison does not at all imply possibility of humans building up resistance. Moongooses (Mongeese?) are unaffected by snake's neurotoxin because of unique shape of receptors to which $\alpha$-neurotoxin is supposed to bind leaving them unable to operate; snake's neurotoxin can not attach itself to those receptors. It's a specific adaptation to hunting snakes (snakes also have unusual shape of those receptors, just to be clear, so snakes are also unaffected). You have no way of transferring modified neuroreceptors to humans, outside of slow mode called evolution (unless your world is in beyond ours genetic engineering phase) which could lead to so called convergent evolution giving some humans complete, inheritable immunity.
According to research dating back to 80s newborn's immune response is "immature" not due to lack of antibodies, but due to lack of antigen (think poison or pathogen, antigens are things that antibodies bind to to tag or neutralise them) stimulation. Summary of paper also states that late-term foetus and newborn baby don't rely exclusively on antibodies "given" by mother, newborn's blood contains antibodies designed to fight mother's lymphocytes, remember that cells themselves are not supposed to pass placental barrier between mother and fetus so ability to secrete such antibodies should be beneficial. Think of immune system not knowing what exactly to fight. Diseases and poisons can hinder development of fetus in many different ways so mother's body becomes extremely zealous about fighting anything that could threaten child-in-making. As such, it is plausible that fetus exposed to your biological poison (poison would have to pass from mother through placental barrier) could start "training" resistance to it, assuming that poison doesn't kill it. That still leaves risk of severe disorders (especially in case of poisons that take very long time to metabolise). Teratology has a long list of disorders caused by toxicants (Drugs in pregnancy, Environmental toxicants and fetal development) ranging from low birth mass through cognitive impairment to death. Famous case was Thalidomide which caused children to be born without limbs, if drug (as in drug the medicine, Thalidomide is used in treating some types of cancer) was taken in early stage of pregnancy.
Extracting antibodies out of blood to create antivenom would be possible, however it's a pretty advanced science (very late XIX century to very early XX century) so plausibility of this part depends on details of your world. Note however that such antivenom will do nothing to help someone already affected, it can only help prevent sedation or death if given fast enough after exposure to poison.
Summary and suggestions
Now, where do we go with this? As I said, fetus building up resistance if mother is poisoned with this poison and poison can pass placental barrier is plausible. This is however immunity in medical sense of being able to resist poison, not immunity in common sense of being unaffected, that requires having different structure of cellular receptors like above mentioned Mongoose. Depending on your world building, you can use sideeffects of prenatal immunisation to include some bonus flavour (or work towards some social/political statement). For example you can create society which is routinely exposed to your poisonous plant and all members of said society are highly resistant but suffer some disorder (your choice, you can make it almost anything, from unusual eye colour though increased paranoia to missing right arm), for ancient-to-feudal-level world you can have nobility valuing resistance to poisons but suffering from, say, increased aggressiveness or diminished intelligence, you can have families of people working with this plant (because it has some other economically important characteristic, e.g. makes good fabrics) who are stigmatised due to abnormally long arms.