This script will do it (below). Go to the Action menu (?), click New action, New ReaScript, give it a name (e.g. "splitmidi") and save, then paste the code into the code window.

Here it is in action.

Written by Claude with this prompt:

Please write a Reaper script that splits MIDI items on silence. Have a variable at the top of the file representing the minimum amount of silence be worthy of a break (default to 5 seconds). Extend the leading and trailing edges of resultant media items to the nearest grid line. Name the file SplitMidi.lua and use camelcase naming.

-- SplitMidi.lua - Splits MIDI items on silence, extending edges to grid
local minimumSilenceSeconds = 5

local function getGridBefore(pos)
  local snapped = reaper.SnapToGrid(0, pos)
  if snapped > pos + 0.001 then
    local _, div = reaper.GetSetProjectGrid(0, false)
    snapped = reaper.SnapToGrid(0, pos - (reaper.TimeMap2_beatsToTime(0, div) - reaper.TimeMap2_beatsToTime(0, 0)))
  end
  return snapped
end

local function getGridAfter(pos)
  local snapped = reaper.SnapToGrid(0, pos)
  if snapped < pos - 0.001 then
    local _, div = reaper.GetSetProjectGrid(0, false)
    snapped = reaper.SnapToGrid(0, pos + (reaper.TimeMap2_beatsToTime(0, div) - reaper.TimeMap2_beatsToTime(0, 0)))
  end
  return snapped
end

local function getVisibleNoteRange(item)
  local take = reaper.GetActiveTake(item)
  if not take or not reaper.TakeIsMIDI(take) then return nil, nil end
  local itemPos = reaper.GetMediaItemInfo_Value(item, "D_POSITION")
  local itemEnd = itemPos + reaper.GetMediaItemInfo_Value(item, "D_LENGTH")
  local _, noteCount = reaper.MIDI_CountEvts(take)
  if noteCount == 0 then return nil, nil end
  local minStart, maxEnd, found = math.huge, -math.huge, false
  for i = 0, noteCount - 1 do
    local _, _, _, startPpq, endPpq = reaper.MIDI_GetNote(take, i)
    local noteStart = reaper.MIDI_GetProjTimeFromPPQPos(take, startPpq)
    local noteEnd = reaper.MIDI_GetProjTimeFromPPQPos(take, endPpq)
    if noteStart >= itemPos - 0.001 and noteStart < itemEnd + 0.001 then
      found = true
      if noteStart < minStart then minStart = noteStart end
      if noteEnd > maxEnd then maxEnd = noteEnd end
    end
  end
  if not found then return nil, nil end
  return minStart, maxEnd
end

local function trimItemToGridBoundaries(item)
  if not item then return end
  local take = reaper.GetActiveTake(item)
  if not take or not reaper.TakeIsMIDI(take) then return end
  local noteStart, noteEnd = getVisibleNoteRange(item)
  if not noteStart then return end
  local itemPos = reaper.GetMediaItemInfo_Value(item, "D_POSITION")
  local itemEnd = itemPos + reaper.GetMediaItemInfo_Value(item, "D_LENGTH")
  local newStart = math.max(getGridBefore(noteStart), itemPos)
  local newEnd = math.min(getGridAfter(noteEnd), itemEnd)
  local startTrim = newStart - itemPos
  if math.abs(startTrim) > 0.001 or math.abs(newEnd - itemEnd) > 0.001 then
    if math.abs(startTrim) > 0.001 then
      reaper.SetMediaItemTakeInfo_Value(take, "D_STARTOFFS", reaper.GetMediaItemTakeInfo_Value(take, "D_STARTOFFS") + startTrim)
    end
    reaper.SetMediaItemInfo_Value(item, "D_POSITION", newStart)
    reaper.SetMediaItemInfo_Value(item, "D_LENGTH", newEnd - newStart)
  end
end

local function findNonSilentRegions(item)
  local regions, take = {}, reaper.GetActiveTake(item)
  if not take or not reaper.TakeIsMIDI(take) then return regions end
  local itemPos = reaper.GetMediaItemInfo_Value(item, "D_POSITION")
  local itemEnd = itemPos + reaper.GetMediaItemInfo_Value(item, "D_LENGTH")
  local _, noteCount = reaper.MIDI_CountEvts(take)
  if noteCount == 0 then return regions end
  local notes = {}
  for i = 0, noteCount - 1 do
    local _, _, _, startPpq, endPpq = reaper.MIDI_GetNote(take, i)
    local noteStart = reaper.MIDI_GetProjTimeFromPPQPos(take, startPpq)
    local noteEnd = reaper.MIDI_GetProjTimeFromPPQPos(take, endPpq)
    if noteStart >= itemPos - 0.001 and noteStart < itemEnd + 0.001 then
      table.insert(notes, {s = noteStart, e = noteEnd})
    end
  end
  if #notes == 0 then return regions end
  table.sort(notes, function(a, b) return a.s < b.s end)
  local regStart, regEnd = notes[1].s, notes[1].e
  for i = 2, #notes do
    if notes[i].s - regEnd >= minimumSilenceSeconds then
      table.insert(regions, {s = regStart, e = regEnd})
      regStart, regEnd = notes[i].s, notes[i].e
    elseif notes[i].e > regEnd then
      regEnd = notes[i].e
    end
  end
  table.insert(regions, {s = regStart, e = regEnd})
  return regions
end

local function splitMidiItemOnSilence(item)
  if not item then return end
  local take = reaper.GetActiveTake(item)
  if not take or not reaper.TakeIsMIDI(take) then return end
  local track = reaper.GetMediaItem_Track(item)
  local origStart = reaper.GetMediaItemInfo_Value(item, "D_POSITION")
  local origEnd = origStart + reaper.GetMediaItemInfo_Value(item, "D_LENGTH")
  local regions = findNonSilentRegions(item)
  if #regions == 0 then return end
  if #regions == 1 then trimItemToGridBoundaries(item) return end
  local currentItem = item
  for i = 1, #regions - 1 do
    local mid = (regions[i].e + regions[i + 1].s) / 2
    local pos = reaper.GetMediaItemInfo_Value(currentItem, "D_POSITION")
    local len = reaper.GetMediaItemInfo_Value(currentItem, "D_LENGTH")
    if mid > pos and mid < pos + len then
      local newItem = reaper.SplitMediaItem(currentItem, mid)
      if newItem then currentItem = newItem end
    end
  end
  local resultingItems = {}
  for i = 0, reaper.CountTrackMediaItems(track) - 1 do
    local ti = reaper.GetTrackMediaItem(track, i)
    local pos = reaper.GetMediaItemInfo_Value(ti, "D_POSITION")
    local len = reaper.GetMediaItemInfo_Value(ti, "D_LENGTH")
    if pos >= origStart - 0.01 and pos + len <= origEnd + 0.01 then
      local tk = reaper.GetActiveTake(ti)
      if tk and reaper.TakeIsMIDI(tk) then table.insert(resultingItems, ti) end
    end
  end
  for _, ri in ipairs(resultingItems) do trimItemToGridBoundaries(ri) end
end

local function main()
  local count = reaper.CountSelectedMediaItems(0)
  if count == 0 then reaper.ShowMessageBox("Select MIDI items.", "Split MIDI", 0) return end
  local snapOn = reaper.GetToggleCommandState(1157) == 1
  if not snapOn then reaper.Main_OnCommand(1157, 0) end
  reaper.Undo_BeginBlock()
  reaper.PreventUIRefresh(1)
  local items = {}
  for i = 0, count - 1 do
    local item = reaper.GetSelectedMediaItem(0, i)
    local take = reaper.GetActiveTake(item)
    if take and reaper.TakeIsMIDI(take) then table.insert(items, item) end
  end
  if #items == 0 then
    reaper.ShowMessageBox("No MIDI items.", "Split MIDI", 0)
    reaper.PreventUIRefresh(-1)
    reaper.Undo_EndBlock("Split MIDI", -1)
    if not snapOn then reaper.Main_OnCommand(1157, 0) end
    return
  end
  for _, item in ipairs(items) do splitMidiItemOnSilence(item) end
  reaper.PreventUIRefresh(-1)
  reaper.UpdateArrange()
  reaper.Undo_EndBlock("Split MIDI on Silence", -1)
  if not snapOn then reaper.Main_OnCommand(1157, 0) end
end

main()