Vietnam can be divided into three regions, namely the north, central and south. Every region has its own weather type, but the southwest monsoon brings rain all over the country. This runs from April to September.
The rainy season in North Vietnam roughly runs from May to October, with July and August having the most precipitation. Showers are usually of short duration and it is nice and warm so no problem.
The rainy season in Central Vietnam roughly runs from September to December, but January can also be very wet. Conditions can certainly be far from ideal in this region and weakened typhoons from the Philippines can make landfall here, resulting in flooding.
The rainy season in South Vietnam roughly runs from May to November, with September and October having the most precipitation. In general, the showers during the rainy season are short-lived and will not disrupt your plans. For a beach holiday you may not be guaranteed a clear blue sky every day, but there are plenty of beautiful days.
Now that you know when the rainy season falls in the different regions, you can hopefully better plan your trip and route. My experience is that you can travel the entire country during the rainy season.
The only thing I would try to avoid is the rainy season in Central Vietnam. It can really be haunted there and really spoil the fun. If you still want to go there, which I can understand with an attraction such as the special Hoi An, take a look at the situation at that moment. Just commit a little less in advance.
If you want to go to the islands of Phu Quoc or Con Dao in the south during the rainy season, you will have to take the showers for granted. You will find much better conditions to chill out in the Nha Trang region. Due to the microclimate in this region, there is considerably less rainfall, which makes a beach like Doc Let the best choice.
Long treks through the jungle or through the mountains of Sapa/Ha Giang can also continue. But you will understand that you will get dirty and that simple sports shoes without a good profile are not ideal for you. Good shoes and rain gear are therefore a must.