I have recently become self-employed (independent contractor) and I would like to take care of my Estimated Taxes as I am paid, which is only a monthly basis. I understand that the IRS requires self-employed people to pay Estimated Taxes (1040-ES) on a quarterly basis so I just want to know if I can pay the 1040-ES on a monthly basis, to avoid any late fees.
No, you can't. Form 1040-ES is supposed to be sent quarterly, per given deadlines. As @Joe said, you can send checks in addition, but that will only cause problems as they won't have the data to match it to and will send you notices of extra money on the account all the time. Or lose it or apply it to something else, or just ignore it because they don't expect it.
Decide what your "withholding" rate should be so that you'd send equal quarterly payments with 1040-ES, put it aside on a savings account, and send quarterly as required. You then will be the one to earn the interest, not the US government.
Strictly speaking, the 1040-ES payments don't have to be equal as @littleadv says emphatically, though making them equal makes it easier to get to "safe harbors" that avoid penalties for not paying enough estimated tax. For example, if your filing status is MFJ and your AGI for this year is less $150K, then if you pay 100% of last year's tax liability in 4 equal timely installments with Form 1040-ES, then there are no penalties for late payment or interest, even if the current year's tax liability works out to be much more than the total amount paid with Form 1040-ES. For high earners (AGI more than $150K), the safe harbor is 110% of last year's tax liability (in timely fashion etc). If you will "make" exactly as much money as a self-employed person this year as you did last year as an employee, you will find this year's tax due number on Form 1040 to work out to be more than last year's because you will also owe both your share of Social Security tax and Medicare tax as well as the employer's share of these taxes. So the 100% safe harbor (or 110% safe harbor) will likely be a good choice to use. Of course, you will have to pay the balance due by next April 15 and so it is a very good idea to put away money for that in a savings account.
There are also other safe harbors that can be used to avoid penalties and interest but they require careful calculations and might give numbers that turn out to be incorrect in view of later developments. For example, large dividends an capital gains from mutual funds which you will typically not know till December, or a considerable increase in self-employment income if you get a new and bigger contract, can cause your current year income to be substantially more than what you anticipated it to be when you filed your first Form 1040-ES in April. So, I recommend the "110% of last year's tax" safe harbor as the simplest to use and one that is guaranteed to work regardless of fluctuations in income.
The 1040-ES is the form to use. It has 4 'deposit tickets' meant for the quarterly payments. From my experience (i.e. unsubstantiated gut feel) it's best to stick with the rules and no more.
Can you? Sure. If you send 52 checks, they'll apply them to your account. Should you? No. It's a waste of the check, paper, stamp and your time. Mark your calendar, set your iPhone reminder, put a sticky note on the door to the garage. Make it a routine.