Yes, you can contribute to a traditional IRA, not take the tax deduction (because you choose not to or because you are income limited) and then convert the traditional IRA to a Roth IRA. The only risk/caveat is that you will get a form 8606 to document the transaction, and you should keep those forms 8606s until you retire, rather than throwing them out after 3 or 6 years.
There is no particular time requirement to do the conversion. If you make a traditional IRA contribution for 2024 in 2024, and then make another traditional IRA contribution in 2025 for 2025, and wait to do the conversion until both contributions have been made, that's fine. Or you can contribute, then convert right away, then contribute again, and convert right away. It doesn't really matter.
The only difference is that any growth that happens in the traditional IRA will be taxable when you convert it. For example, if you contribute $7000 in December, and $7000 in January, and you have $14,200 to convert in February, the $200 is taxable income (because it was growing tax free, it has to be taxed when it is converted to the Roth). For this reason, most people do the transfer right away, and hold the money in a cash account for the couple of days it takes to work through the system. (By contrast, if you contribute $7000 on December 15, convert it on December 16, then contribute $7000 on January 5, and convert it on January 6--then when you have $14,200 in February, that will be tax-free growth because it's in the Roth already.)
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