No - for herd immunity to work, it requires that immunity - whether naturally or vaccine-acquired, stop transmission somewhere around 95% of the time (or more). This is known as sterilizing immunity.
As has become glaringly obvious however, neither having CoVid-19 nor being vaccinated against it, gives rise to sterilizing immunity.
As Khunper pointed out, it is much more likely to be like colds and flu where relatively large proportions of the population will get infected every year - and possibly even several times a year.
Any chance of herd immunity started to recede with the Delta variant, and the chances have only diminished further with subsequent variants.
Now, with the increased transmissibility and immune evasiveness of Omicron and especially the BA.4 and BA.5 variants, the prospect of herd immunity is really only a distant memory.