"has started posting" is correct and usable in written English.
Generally speaking, it's a good fit for situations when something has recently begun posting, such as an individual or business. For example: "Our company has started posting content twice a week to stay in touch with our customers."
Zero-Knowledge, for example, has started posting essays under the heading Demystifying Cryptography on its Web site.
The insurer has started posting voting packs to members that include a share allocation statement and a voting form.
Un-Post My Heart I'm on frigid terms with my ex-boyfriend, who has started posting radical views on his blog.
Since January, Apple has released the names of 156 of its suppliers — which it had previously declined to identify — and has started posting regular monitoring reports on the number of hours worked by factory employees.
Without any fanfare, the House Oversight and Government Reform Committee has started posting preliminary transcripts of many of its hearings on its Web site, giving everyone a chance to pore through testimony and find news the MSM may have overlooked.
I probably started on 2 December, because friends had started posting pictures of their trees online.
In early January 2012, I noticed that another Jon Ronson had started posting on Twitter.
For instance: should the editor have started posting live updates to Twitter of his conversations with the hostage-taker?
A spokesperson from Simply Be suggested this was because celebrities have started posting selfies of themselves in pyjamas.
To raise money, the men have started posting themselves on street corners with orange collection buckets, asking passers-by for change.
It's reported members of Isis in command positions have started posting propaganda saying that the group has its eye on the West, an ominous sign given the trouble we are having dealing with them in Iraq and Syria.
