Edited for clarity.
From what I can see online the chuck has a JT33 taper, didn't see any info on a spindle taper but that doesn't matter. You should be able to remove the chuck with a pry bar or a couple gentle taps on the top using a hammer and brass drift. Using the hammer and drift method, tap, turn a bit, tap and turn, tap being the ultimate word, don't get heavy handed..
Your choice, but I doubt you will be happy with a keyless chuck in a drill press. When shopping for a chuck be aware that there are some that only close up so far and will not hold small bits.

 

Thanks for responding. From what I've seen isn't there supposed to be an open slot in the drill press shaft to allow the chuck to be removed if it is a morris taper chuck?

 

Based on this illustration the chuck is held on by a taper.

https://www.ereplacementparts.com/spindle-shaft-p-151331.html

Even if you can remove that chuck, you'll need to find one with the same taper.
There's a tool called a "pickle fork" used to separate tapered parts on cars and trucks.
It's a wedge with a slot in the center which you drive in between the chuck and the spindle to force them apart.
You may be able to rent one from an auto parts store like O Reilly's.
A good mechanic will have one you can borrow.
A hardwood version may also work if you can make one.
A good sized screw driver may also work.

 

Jacobs taper?

I don’t care for keyless chucks. Are the teeth in good condition on the wrench? They tend to wear over time. Also check the spacing between the teeth. The chucks can be disassembled. They are pressed together. The outer sleeve with the teeth has to be pressed to the correct position so the teeth mesh properly.

 

Drill chuck removal wedges are designed specifically to this task. They are designed to be used as a pair, which provides a straight on centerline force.

The wedges are available in several sizes. You need to use the size appropriate for the drill chuck you are removing.
Notice in the photos it shows sizes#1, #2, and #3. I believe the wedges go up to a size #8.

 

There are usually three parts involved, the spindle, the arbor and the chuck. In your case it appears there is not an arbor, the end of the spindle has a Jacobs taper to attach the chuck. You need to remove the chuck from that short taper.
Did you watch the video linked to in post #4?

 

From what I've seen isn't there supposed to be an open slot in the drill press shaft to allow the chuck to be removed if it is a morris taper chuck?

Your drill press does not use a morse taper to secure the drill chuck.
The taper used has been machined directly onto the spindle as a male taper. The chuck has a matching female taper.
Drill chuck removal wedges are designed specifically to this task.

 

Could you please explain how to remove the chuck and give me an option or 2 for the keyless chuck that would work in this drill press?

You have been given options for chuck removal. Once the chuck is removed, you will need to take precise measurements so as to determine the style and size of taper used. Then you will need to find a chuck which offers a matching taper. No one can recommend a chuck until the taper style and size has been determined.

 

@BGST

to put on

Same Chucky style that Craftsman used many a year ago and it used an odd key point which causes not to work correctly if you use a different one because it used to collard insert on the stem.
So standard Chuck keys will not work

 

This chuck should work from the description details:

Nakkaa 1/32"- 5/8" MT2 Shank Heavy Duty JT33 Keyless Drill Chuck Compatible with Lathe Drilling Machine - Amazon.com

Nakkaa 1/32"- 5/8" MT2 Shank Heavy Duty JT33 Keyless Drill Chuck Compatible with Lathe Drilling Machine - Amazon.com

www.amazon.com

 

Why pay extra for the arbor that is not used, cheaper to just purchase a chuck alone.

 

So, post the link.
Did you open the entire page?
They all come with the arbors .... $26 ain't all that bad .

 

 

That's the good thing about Amazon, you can see the reviews.
Almost everything is made in China these days, so no expectations of great quality.
For real made in USA quality you'll pay a lot more:

Pardon Our Interruption...

 

Good luck finding a Jacobs chuck made in USA, been manufactured off shore since 2010 and apparently not up to former standards.

 

PLEASE don't hate on me but,this took less than a minute to find. I don't care what anyone says,or feels about the "bay".... it is invaluable for realtime data. Meaning,not only on availability,but current cost comparison.

This seller lists a tech phone number even. No affiliation but can tell in his pricing and general fung shway that he's "serious".

Best of luck with your project. Oh yeah,use opposing wedges as posted above for proper removal.... I'd still call Ebay boy about it though,he may have a "hack"?