Is it possible to calculate p-values for fold changes of single replicate RNA-seq samples?
DESeq2 is able to adjust p-values for single-replicate samples by estimating shared dispersion across all samples. DESeq2 will give a warning, but try its best to carry out the analysis. More information on doing that can be found in the DESeq2 vignette and the DESeq2 manual page.

The results will not be anywhere near the sensitivity that can be achieved when replicate information is available, but it might be good enough if only very obvious large-scale changes are of importance.