Keep track yourself with your own records.
Contributions are reported to you by the broker on form 5498, usually issued in May of the following year. The IRS gets a copy of form 5498, but they only keep copies for 7 years and they don't keep track for you.
The broker will not keep track for you, for various reasons that I can explain if you are really curious.
IRA means individual retirement arrangement. You only have one arrangement, regardless of how many different accounts you have at different brokers. Your contributions and withdrawals are combined for tax purposes. (If you contribute $5000 to account A and $5000 to account B, and then withdraw $6000 from account A, that counts as all basis, and you have $4000 of basis left in your IRA even though it is not balanced across accounts.) So you only need to keep track of your total lifetime contributions, total lifetime conversions, and total lifetime withdrawals--you don't have to track multiple accounts separately if you don't want to.
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