Which Vancouver? Amtrak actually has Vancouver, BC and Vancouver, WA on the Cascades route.
Silver Service does have a baggage car and checked baggage service. It may say "no checked bags" on your itinerary, but it's because Palatka is unstaffed. Usually - only station staff is allowed to handle baggage; there's no baggage handlers on the train. On top of that baggage may just go to the destination and be held there. They certainly don't want to leave the baggage on the platform, and if they went to a trackside claims system, they have a mess if one goes unclaimed. Amtrak has some experimental trackside check-in (departure only) at certain stations, but I don't think Palatka is one of them.
You can look at the schedule, and there's a little icon for a piece of luggage for every station that has baggage service. The stations just before and after Palatka both have baggage service. I've heard that you could theoretically ask to get it checked in to Jacksonville if you have some way to pick it up.
https://www.amtrak.com/content/dam/...s/Silver-Service-Palmetto-Schedule-110216.pdf
They don't really need more than one station employee to handle baggage at smaller staffed stations. I was at a station early and chatted with the single Amtrak employee working at that station. She said it was only two employees - each 5 days a week with overlaps most days. If she was working alone, she was responsible for everything including ticketing, announcements, and baggage handling.
I've heard of some bags checked in for a really short trip - like Los Angeles to Van Nuys. The rationale I heard for that was to avoid a baggage storage fee, where there's that 24 hours to claim a piece. There are some really short trips where baggage could theoretically be checked, such as Emeryville to Oakland (5 minutes), or Anaheim to Santa Ana (8 minutes).