The short answer is no – not if your main goal is to experience Vejle itself. The centre is compact, the railway station is conveniently placed and walking is often the easiest way to move between shops, restaurants and central sights.
Vejle without a car
For a weekend focused on the city, I would happily arrive by train and leave the car behind. You can walk from the centre towards the harbour and reach landmarks such as Fjordenhus without turning transportation into part of the planning.
Local buses and trains can also help with selected trips beyond the centre. Jelling, for example, is a straightforward destination to consider without driving.
When a car becomes useful
A car starts making more sense when the surrounding landscape becomes a major part of your itinerary. If you want to explore smaller places around Vejle Fjord, countryside attractions or combine several destinations in one day, driving gives you much more freedom.
It can also be practical for families carrying plenty of luggage or travellers staying outside central Vejle.
If You plan a day trip to Legoland, I recommend go by car.
Choose based on the trip
I would not rent a car merely because Vejle is a smaller city. Decide what you actually want to see first. For a city break, walking and public transport may be simpler. For a wider tour of the region, a car can transform what is realistically accessible.
That flexibility is one of Vejle’s strengths: it works both as an easy rail destination and as a starting point for a road trip through this part of Denmark.