Is it a sin to go to the mosque with body odor?

No, it isn't a sin if you can't help it, but you should do what you reasonably can. The Prophet ﷺ taught Muslims to come to prayer clean and pleasant, because strong smells bother worshippers standing beside you. If you simply didn't have time or have a medical reason, don't stay away. If you could easily have freshened up, it's better to do so.

What the Prophet ﷺ taught

About raw garlic and onion, the Prophet ﷺ said that whoever eats them must not approach the mosque, because "the angels are harmed by the same things as men" (Sahih Muslim 564). Scholars apply this reasoning to any strong smell that bothers others, such as heavy sweat or dirty socks. This isn't about shaming anyone. It's about courtesy for the people next to you, shoulder to shoulder in the row. See Can I eat garlic before going to the mosque?

The Quran also asks us to come prepared:

يَـٰبَنِىٓ ءَادَمَ خُذُوا۟ زِينَتَكُمْ عِندَ كُلِّ مَسْجِدٍ "O Children of Adam! Dress properly whenever you are at worship." Quran 7:31

The story of the sweaty workers

Aisha reported that people used to come to Jumu'ah from the outskirts of Madinah, covered in dust and dripping with sweat from the walk. One of them came to the Prophet ﷺ, who said, "I wish that you keep yourself clean on this day" (Sahih al-Bukhari 902). He didn't tell them to stay home. He encouraged them to clean up, with kindness. That's the right spirit for all of us.

Is it actually a sin?

Not if you can't help it: a medical condition, medication or a long day that left no time for a shower is not a sin. Allah doesn't hold you responsible for what's beyond your control.
Disliked (makruh) if you ignore it: if you could easily wash, change or use deodorant and chose not to, scholars call it disliked, because it troubles other worshippers.
Never a reason to skip prayer: a prayer in the mosque with a little sweat is better than missing it. Just do your best to tidy up first.

Simple steps before the mosque

Shower or wash underarms and feet, especially after exercise or work.
Change your shirt and socks. Fabric holds odours, so a spare set in the car or bag is very useful.
Use deodorant, and keep any fragrance light. Too much perfume bothers people too.
Take care of your mouth: brush, use a miswak and avoid strong foods beforehand.
Wudu helps: washing the face, hands and feet cleans the places that smell most.

Friday is the best occasion to do it all. The Prophet ﷺ encouraged a bath, the miswak and perfume before Jumu'ah (Sunan Abi Dawud 344). See What are the sunnahs of Friday?

If you struggle with body odour

Some people sweat heavily or have a persistent smell despite washing. That can be a medical issue, and a doctor can often help with prescription antiperspirants or treatment. Don't be embarrassed to ask. If you're fasting, deodorant is fine. See Can I wear deodorant while fasting?

Key points

Strong smells bother other worshippers, so come clean and fresh.
It isn't a sin if you couldn't avoid it.
Never skip prayer because of embarrassment.
A spare shirt, deodorant and a miswak solve most problems.

Tip: Keep a small "mosque kit" in your bag or car: a spare pair of socks, a travel deodorant, a miswak and a clean top. It takes thirty seconds to freshen up.