My company docks Paid Time Off or Vacation time automatically if someone is late to work. They do not give an option of using our lunch time (working through it to "pay back the company"). They consider it as this - If an employee is late then they are taking the company's time so they will take our vacation time. The payroll company we use wants everything to be nice and easy and always show 40 hours for everybody which is why they automatically take our vacation time away. I thought that if they give us vacation it is up to us as to how to use that time. I know they can limit the number of days or periods (seasonal highs) that vacation can be taken but that is my time that I earned by working here and if I am late then I either don't get paid or should have the option to use whatever time I have (whether it is lunch, PTO, or vacation).
They also do not pay for the hours that we work. Hours of Operation are from 9:00am to 6:00pm with a 1 hour lunch that is a set time. We use a time clock machine and even if my lunch is 12:00 to 1:00 and I clock out at 12:02 and back in at 12:58 then clock out at the end of the day at 6:05 I only get paid 8 hours not the additional time. Overtime is not authorized unless we are on late issues and it has been approved. I thought that as an hourly employee they have to pay us for the time we work regardless.

Is any of this legal?

This is a duplicate post but since you have provided additional information I'll answer it anyway.

I thought that if they give us vacation it is up to us as to how to use that time.

You were wrong. The company does not have to give you vacation time; they get to set the rules about how it is used. If you don't want your vacation time deducted to cover a tardy, then don't come in late. This is perfectly legal.

Rounding is permissable as long as it operates in both directions.

It is not illegal in your state for the employer to dock your vacation pay when you are late. The employer does not have to offer paid time off at all, and the company can set whatever criteria it chooses on how it is used. It is not true that vacation usage is at the whim of the employee. The resolution for this is not to be late.

It is also not illegal for the employer to round your in/out times. FLSA regulations are that, generally speaking, times may be rounded to the nearest 15 minutes and that, as long as the rounding is done consistently and results in the employee being paid "substantially" for time worked, that is acceptable. Accordingly, the rounding rules being used at your company, based on your example, are within the regulations of the federal Fair Labor Standards Act.

So basically everyone here is wrong in thinking that when we earn vacation time, it is not ours. I can understand it if we were salaried employees and we were essentially "stealing" time from the company, but since we are hourly we should be paid for the time worked and if we are late we don't get paid for that time. I understand rounding but does that happen at clock in/out or at the end of the week?

I'll let Patty answer the rounding question.

If your belief is that you have 100% freedom to take vacation only as and when you want and your employer cannot require you to take it or not take it at any given time, yes, you are wrong.

I say again, if you don't like your vacation being docked when you come in late, don't come in late.

As I said, rounding to the nearest 15 minutes is acceptable, as long as it is done consistently, no matter to whose benefit the rounding accrues. And needs to be done daily.

So, let's look at an allowable example.

Clock in = 9:05; round to 9 (in lieu of using vacation)
Out for lunch = 12:02; round to 12:00
In from lunch = 12:58; round to 1:00
Clock out = 6:10; round to 6:15

Now, if they rounded to 6:00 when you clocked out at 6:10, that would not be allowed, unless, if you clocked in at 9:10, they also rounded down to 9:00. I do not see any violations of wage and hour law in the way they are applying the rounding rules.