Short answer
No. The sunnah is to wear your best clothes on Eid, new or not. Umar once suggested the Prophet ﷺ buy a cloak to look his best for Eid, which shows dressing up for Eid was their practice. The Hanafi texts list wearing one's best clothes among the sunnahs of the day. New clothes are a lovely custom if you can afford them, but not required. The four schools agree.
On Eid it is mustahabb to bathe, use the siwak, wear perfume and put on one's best clothes, whether new or washed. Buying new clothes is not required, and men still avoid silk and gold.
The four schools agree on this.
Explanation
Umar found a silk cloak for sale and brought it to the Prophet ﷺ, suggesting he buy it to look his best for Eid and for receiving delegations. The Prophet declined only because it was silk, which he said is the dress of those with no share in the Hereafter. So dressing well for Eid was normal; the problem was the fabric.
The sunnah. The Hanafi books list among the sunnahs of Eid: ghusl, the siwak, perfume and wearing your best clothes. "Best" means the best you have. If you already own a good outfit, wash it and wear it. New clothes are a joy, especially for children, but there is no sin in not buying any. See Eid and how to pray Eid salah.
For men. No silk or gold, trousers above the ankles, and the usual dress code; see the dress code for men. White is popular, but any good clothes are fine; see wearing white.
For women. Wear your best at home and in women's gatherings. When going out to the Eid prayer or visiting, choose clothes that cover properly and do not attract attention, and skip strong perfume; see the hijab guide.
Do not overspend. The Prophet said Allah loves to see the signs of His blessing on His servant, but going into debt or wasting money on Eid outfits goes against the spirit of the day.
The evidence
Quran, Surah Al-A'raaf 7:31
۞ يَٰبَنِىٓ ءَادَمَ خُذُوا۟ زِينَتَكُمْ عِندَ كُلِّ مَسْجِدٍۢ وَكُلُوا۟ وَٱشْرَبُوا۟ وَلَا تُسْرِفُوٓا۟ ۚ إِنَّهُۥ لَا يُحِبُّ ٱلْمُسْرِفِينَ ﴿٣١﴾
O children of Adam, take your adornment at every masjid, and eat and drink, but be not excessive. Indeed, He likes not those who commit excess.
Dress well for worship without excess.
Sahih al-Bukhari 948
Sahih... ابتع هذه تجمل بها للعيد والوفود ...
"Buy this and beautify yourself with it for Eid and for receiving delegations."
Dressing well for Eid was the known practice.
Jami' at-Tirmidhi 2819
Grading to be checked... إن الله يحب أن يرى أثر نعمته على عبده ...
"Allah loves to see the signs of His blessing on His servant."
Wearing good clothes within one's means is praised.
Where the ruling is recorded
- al-Hidaya, Kitab al-Salah, chapter on the Eid prayers. On the day of Eid one bathes, uses the siwak, applies perfume and wears one's best clothes.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.