This is an old question but it recently [29 Apr 2022] assisted me in finding a solution to a problem I was faced with. To say thanks I combined the approaches of @TonioLiebrand and @JeremyVoisey into a single self-contained (neatly rounded-off) function as follows:
# Self-contained function to calculate 'Leaflet Map Widget' Bounding Box co-ordinates ...
"f.calc.leaflet.bounding.box.coords" <- function(objLeafletMap=NULL) {
  FUNC_ID_SHORT <- "fCLBBC"; FUNC_ID_FULL <- "f.calc.leaflet.bounding.box.coords";
  boundNorth_ <- NULL; boundWest_ <- NULL; boundSouth_ <- NULL; boundEast_ <- NULL;
  if (is.null(objLeafletMap) || is.null(objLeafletMap$width) || is.null(objLeafletMap$height)) {
    base::message(base::paste0(" --> ", FUNC_ID_SHORT, " - Line 4 ", "| LEAFLET MAP WIDTH & HEIGHT NOT SPECIFIED ( FUNC ID: '", FUNC_ID_FULL, "' )."))
    base::stop(base::paste0(" --> ", FUNC_ID_SHORT, " - Line 5 ", " | Function Execution Terminated [ reason: REQUIRED PARAMS are NULL ] !!"))
  } else {        
    # Extract Leaflet Map Widget 'x' property list values ( i.e. [center 'lat' & 
    # 'lon'], 'zoom' and 'options' widget property values )...
    view_ <- objLeafletMap$x$setView; zoom_ <- view_[[2]]; 
    lon_ <- view_[[1]][2]; lat_ <- view_[[1]][1];
    
    # Extract Leaflet Map Widget 'width' and 'height' values ...
    width_ <- objLeafletMap$width; height_ <- objLeafletMap$height;
    
    # Calculate Leaflet Map Widget peripheral extent in co-ordinate dimensions ...
    lon_width_ <- 360 * width_ / 2 ^ (zoom_ + 8);
    lat_height_ <- 360 * height_ * cos(lat_ / 180 * base::pi) / 2 ^ (zoom_ + 8);
    
    # Calculate Leaflet Map Widget peripheral extent ( i.e. Bounding Box ) in co-ordinate values ...
    boundEast_ <- lon_ + lon_width_ / 2; boundWest_ <- lon_ - lon_width_ / 2;
    boundNorth_ <- lat_ + lat_height_ / 2; boundSouth_ <- lat_ - lat_height_ / 2;
  }
  return(list(top=boundNorth_, right=boundEast_, bottom=boundSouth_, left=boundWest_, zoom=zoom_))
}
This might be a bit of an overkill for some, but might also be a boon for others looking for a quick [time-critical] solution (as I did). Simply copy & paste the function into your R script, and once the function is read into your R Session Memory extract your Leaflet Map Widget bounding co-ordinates as follows:
# Simply call the function from wherever you need in your R script ...
mapBounds <- f.calc.leaflet.bounding.box.coords(m);   # <- 'm' == Leaflet Map Widget (with 'width' and 'height' defined) !!
# ... and extract the results as follows:
mapBounds$top
> -5.83050217387398
mapBounds$right
> 38.25046875 
mapBounds$bottom
> -40.209497826126 
mapBounds$left
> -9.21046875
I also added the zoom value as an output (because it seems wasteful to compute it inside the function but not return it [as a result] to the function call). So now you can easily easily get the zoom value by calling ...
mapBounds$zoom
> 5
... after every map zoom change - if you really, really need to do that.
Lastly, I concur with @JeremyVoisey, there is an accuracy issue with the results from this approach - but this code snippet was sufficient in resolving the problem I had (and I was a bit time-pressed) ... so I didn't look into fixing the accuracy issue at the time.