My guess is that there's a bell curve and that approximately 25-35% is ineligible at any time and maybe 10-25% is perpetually ineligible while 10-25% is perpetually eligible. That's just a complete guess.
1) You must be a legal resident of Texas for the six month period preceding the date of application.
That would disqualify a small chunk.
2) You must be at least 21 years of age. a person who is at least 18 years of age but not yet 21 years of age is eligible for a license to carry a concealed handgun if the person: (1) is a member or veteran of the United States armed forces, including a member or veteran of the reserves or national guard: (2) was discharged under honorable conditions, if discharged from the United States armed forces, reserves, or national guard: and (3) meets the other eligibility requirements of Subsection (a) except for the minimum age required by federal law to purchase a handgun.
It's easy enough to figure out the number of people older than 21 and younger than 21.
3) You must not have been convicted of a felony, including with probation or deferred adjudication.
The number of felony convictions is easy enough to determine.
4) You must not be currently charged with the commission of a felony, a Class A or B misdemeanor, or an offense under Section 42.01 of the Penal Code (Disorderly Conduct).
I'd guess that's a decent chunk. The courts stay pretty busy and it isn't all Class C.
5) You must not be a fugitive from justice for a felony or a Class A or B misdemeanor.
We know there are a lot of warrants out there. It would be interesting to see just how many. I seem to recall a news piece talking about how many one time. I wouldn't be surprised if the number was available in some report.
6) You must not be a chemically dependent person. A person with two convictions within the ten year period preceding the date of application for offenses (Class B or greater) involving the use of alcohol or a controlled substance is ineligible as a chemically dependent person.
Sadly, I think this is a small chunk of our population.
7) You must not be incapable of exercising sound judgment with respect to the proper use and storage of a handgun.
Few would fall into this category, but you could see how many are declared incompetent or have power of attorney over another adult.
8) You must not have been convicted in the five years before the date of application of a Class A or B misdemeanor or an offense under Section 42.01 of the Penal Code (Disorderly Conduct).
The number of people who fall into this category should be easy enough to figure out.
9) You must be fully qualified under federal and state jaw to purchase a handgun.
I don't have a clue how that affects someone that the others don't.
10) You must not have been finally determined to be delinquent in making a child support payment administered or collected by the attorney general.
The child support enforcement office stays pretty busy. How many have final determinations of delinquency? 
11) You must not have been finally determined to be delinquent in the payment of taxes collected by the comptroller, state treasurer, tax collector of a political subdivision, Alcoholic Beverage Commission or any other agency or subdivision.
Property taxes, court fines, water bills... but not just behind, finally determined to be delinquent.
12) You must not have been finally determined to be in default on a loan made under Ch. 57 of the Education Code.
My husband thinks this might be off the requirements now. In this economy, I wouldn't be surprised if there were a number of people disqualified for this reason.
13) You must not be currently restricted under a protective order or restraining order affecting the spousal relationship.
It's easy enough to see how many of those there are.
14) You must not have been adjudicated as having engaged in delinquent conduct violating a felony-grade law within 10 years of the date of application. This section applies only to juvenile offenses, not to felonies committed by adults. Juveniles are not convicted unless certified as an adult.
Sadly, I think the juvie courts stay busy with young delinquents, too.
15) You must not have made any material misrepresentation or failed to disclose any material fact in an application or request for application.
I doubt that this disqualifies much of anyone.
wha dat mean?