People usually want this for one reason: a video as a Lock Screen wallpaper that moves when you press it. The conversion is real, but it cannot happen on a website. Here is why, and the short path that works.
Why a website cannot do this
A Live Photo is not one file. It is a still image and a three-second video, paired by hidden metadata, and the pair only counts as a Live Photo once it sits inside the iPhone photo library. A browser can hand you files; it cannot install that pair. Any site claiming to convert video to Live Photo ends at a download that arrives as a plain video. The install step belongs to an app on the device.
Step 1: Trim to the moment
Live Photos play about three seconds. Cut your video down to the moment you want before converting - the free trim tool does it in the browser, no watermark.
Step 2: Convert with a Live Photo app
On the iPhone, an app such as intoLive takes a video from your library and saves it back as a genuine Live Photo. The free tier covers clips of a few seconds, which is all a Live Photo holds anyway.
Step 3: Save and set as wallpaper
Save the result to Photos, open Settings, choose it as a Lock Screen wallpaper, and press the screen to see it play.
Going the other way is easier
Live Photo to video needs no app at all: upload the Live Photo's .mov file to the free Live Photo to Video converter and download an MP4 that WhatsApp and Instagram treat as a real video.