Making a pin chuck is easy. Start with a piece of mild steel rod and file a section about one inch long and deep enough so when a nail is laid on the flat it is just as high as the un-filed portion.

I predrill the blank and using the live center to assure the hole is parallel to the ways, I rough turn the blank .

I chuck the round end in my Nova Chuck, lay the nail (pin) on the flat area of the rod. The foam is there so I don't mar the top of the vase

I slip on the vase blank.
Notice the punch marks on the steel rod... they are there so I know where the pin is.
Otherwise when I remove the vase I could lose the small pin in a pile of shavings.

Bring up the tailstock to lightly support the blank.

Turn on the lathe and turn the blank. The outside is turned concentric to the hole.

I turn the bottom and leave a little nib

Which I turn off with a 1/4" flutless gouge

Slip a test tube into the hole and you have a very nice vase ready to use.