I am currently a final year PhD student in Computational Biophysics and I want to explore different opportunities for my future career. I am not sure if tweeting about my job search would be considered bad by potential employers. Is using social media for academic networking and career development appropriate? For further context, I am from India, and could not afford to travel to US or Europe for conferences, which is one way people land Postdoctoral offers. I also do not have many publications (3 first author papers) which is why I believe I have difficulties in getting a position.
I was in a similar situation about a year and a half ago. My condition was worse than yours as I had no accepted publication, only a couple of pre-prints. I submitted 6 applications via job boards and sent 47 emails before I received an offer.
I agree with the commenters -- a tweet is unlikely to help you land a position. But here are a few things that I did that eventually helped me land a position.
If you have a list of potential labs you want to join, follow the PIs on Twitter. If they have a job available, they generally tweet about it (also follow them on Mastodon). Once you start following 15-20 PIs in a subfield, the tweets from their extended network will also appear in your Twitter feed (via retweets, likes, and network effect). So, that might be a good way of knowing if any position is available. Additionally, you can follow these handles on Twitter -- @VacancyEdu, @jobRxiv, @PostdocPal. There might be more, but I used to follow these, and they used to tweet about vacancies regularly.
Check University job portals regularly. Most reputable US and EU universities have separate job boards where they post vacancies. Apply to open job advertisements if you are interested in the position. You are much more likely to be considered for a position if the PI is actively looking for someone than if you cold-email them.
Finally, if you are really interested in working with someone, send a personalized email. Don't send a generic email; they'll see through it and promptly mark it as spam. Don't get disheartened if you don't get positive responses (or any response at all) from PIs. PIs are extremely busy and get about 30-40 emails on an average day. Many of them require their immediate attention, and as much as they would like to (or so I've heard), they can't reply to everyone. If someone doesn't reply, assume that they are not actively looking for someone and move on. Don't take it to mean anything more.
What you're describing is quite common nowadays; i.e., posting a tweet that you are looking for a position. While it may not land you the position you want, I don't see how it can hurt, so why not post the tweet?
Consider the specific culture in your area: are most researchers active on twitter? Or are they more likely to respond positively to a simple email asking to circulate your interest?
It would help a whole lot if you have an advisor or another trusted mentor who will share the tweet and say good things about you. Is your advisor on twitter? You can even ask them to post on your behalf. Alternatively, if people in your discipline know you, and like/retweet etc., that can help a tweet gain traction.
Finally, look for examples of others in your field who have successfully posted tweets with high engagement. What did they write? I have seen many candidates do this very successfully, mostly for faculty positions but I think a similar thing applies to postdocs.
All comments and answers so far basically say that writing a tweet announcing that you're seeking postdoctoral jobs is likely not very effective (unless perhaps if you have a following like this friend of mine with 70,000+ followers on Twitter; she was already a bit of an academic celebrity and getting paid to give talks even in her earliest years of grad school).
However I'll write a more direct answer to your question:
"How appropriate is it to post a tweet saying that I am looking for postdoc positions?"
It will hardly be deemed "inappropriate". It isn't the conventional way of finding a postdoctoral job (which your question's body already indicates you know), but most potential supervisors will not blacklist you for posting this tweet.
"Is using social media for academic networking and career development appropriate?"
In terms of using social media for networking (not just announcing job availability), it is quite common in your field of computational chemistry. Immediately I can think of a lot of computational chemistry professors who are very active on Facebook and Twitter, for example, here's just three off the top of my head that have interacted with me in the past despite my profile only having 71 followers because I have barely ever used Twitter):
If needed, I could give you a list of 30 more, that would probably be very happy to "connect" with you on social media, but this often means nothing in terms of helping you get a postdoctoral job (the people I know have enough people applying through conventional means, and while some people might be known for re-tweeting their own students' job availability posts, they're very unlikely to do it for someone that they don't know because of the risk on their reputation if you turn out to be a very bad postdoc).
Also, in terms of using social media for networking, I'm reminded of the answer I gave to this question: Is it acceptable for a student to connect with a professor on LinkedIn? Out of 13 answers mine was accepted, and got 24 upvotes and 0 downvotes so it might be helpful to you too.