Can the central dogma work in reverse?
Let’s first consider what the Central Dogma[1] actually says. It is precisely summarised in the following figure[2]:


solid arrows represent transfer of information that has been observed directly; dashed arrows represent potential, not yet observed, transfer of information

In the time since the original formulation of the Central Dogma in 1958, we have been able to contribute evidence for one of the dashed lines: RNA→DNA is performed in vivo via reverse transcriptase. By contrast, note the absence of any arrows originating from “protein”. The idea of the Central Dogma was that proteins were an information transfer sink: information could only flow in, not out.
And Crick and his contemporaries had good reasons for assuming such an information sink: translation from RNA to proteins is lossy  because the genetic code is degenerate. A given amino acid can be encoded by more than one codon.
This doesn’t mean that polypeptide chains couldn’t be reverse translated into “a” RNA representation. But it wouldn’t necessarily be the original RNA. Stated like this, it’s obviously possible: we can in fact take a polypeptide chain, sequence it, and synthesise a corresponding RNA.
However, this doesn’t happen in nature — at least, it hasn’t been observed so far, and there are good reasons to assume it doesn’t happen at all. In fact, Crick’s paper[1] explains:

The transfer protein→RNA … would have required (back) translation, that is, the transfer from one alphabet to a structurally quite different one. It was realized that forward translation involved very complex machinery. Moreover, it seemed unlikely on general grounds that this machinery could easily work backwards. The only reasonable alternative was that the cell had evolved an entirely separate set of complicated machinery for back translation, and of this there was no trace, and no reason to believe that it might be needed.

The “complicated machinery” Crick mentions here consists of two distinct parts: On the one hand the ribosome, and on the other hand the aminoacyl tRNA synthetases (aaRSs, singular aaRS). The latter is essentially the physical embodiment of the genetic code in the cell. The former is the machinery necessary to decode RNAs using the codon adapters (= tRNAs) charged by the aaRSs.
Crick correctly notes that, in order to allow the transfer protein→RNA, the cell would require a system analogous to the ribosome and all the aaRSs, redundantly encoding the exact same information. And it would need to ensure that these two distinct encoding systems (translation and reverse translation) never get out of sync with each other.
It’s worth contemplating, briefly, why the cellular translation process isn’t simply reversible, given that it’s an chemical process, and thus pretty much by definition reversible in theory. The reason for this is that translation happens in two entirely distinct, and physically separate, stages: the charging of individual tRNAs with their corresponding amino acids by an aaRS, and the actual mRNA translation by the ribosome.
Simply reversing the ribosome activity would result in a polypeptide being digested from one end, yielding individual amino acids. However, the ribosome actually has no facility for capturing these amino acids, they would simply diffuse away without being attached to a tRNA, let alone the right tRNA. A successful reverse translation would have to physically couple each type of aaRS in turn to the ribosome to ensure that terminal amino acids would pair with the correct tRNA, which, in turn, would pair with a codon. There are two fundamental objections to it, which render it impossible:

Codons, which could be combined into mRNA chains, don’t swim around individually. In fact, short RNA chains are highly unstable and are quickly degraded. By contrast, individual ribonucleotides can be combined into chains — that’s what polymerases do all day, after all. But the ribosome doesn’t possess polymerase activity, and the tRNA anticodon would not provide sufficient physical support to enable a polymerase to dock to it and start synthesising a secondary strand.
Steric hindrance prevents the aaRS enzymes from interacting with the tRNAs while they are being held by the ribosome: the ribosome and the aaRSs would have to physically overlap.

It’s hard to get definitive answers in science purely from theoretical considerations, but this one’s as certain as you’re likely to get: cellular translation using the ribosome is a non-reversible process. Reverse translation, if it existed, would therefore require a separate machinery.