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I don't think this is possible. The usual way is to write and close, then read. If you want to write again, just overwrite the memfile object.

For example:

from rasterio.io import MemoryFile
import rasterio
import numpy as np

def create_memory_file(data, west_bound, north_bound, cellsize, driver='GTIFF'):
    #data is a numpy array
    dtype = data.dtype
    shape = data.shape
    transform = rasterio.transform.from_origin(west_bound, north_bound, cellsize, cellsize)
    with MemoryFile() as memfile:
        with memfile.open(
                driver=driver, width= shape[1], height = shape[0],
                transform=transform, count=1, dtype=dtype) as dataset:
            dataset.write(data, 1)
        return memfile.open()  # <==== Re-open the memfile
  

data = np.array([[1,2,3], [4,5,6], [7,8,9]]).astype(np.int32)
memfile = create_memory_file(data, 0, 2, 0.5)
print(memfile.read())

data = np.array([[4,5,6], [7,8,9], [10,11, 12]]).astype(np.int32)
memfile = create_memory_file(data, 0, 2, 0.5)
print(memfile.read())

Output:

[[[1 2 3]
  [4 5 6]
  [7 8 9]]]

[[[ 4  5  6]
  [ 7  8  9]
  [10 11 12]]]