Using v-model with a prop on VUE.JS

_ prefixed properties are reserved for Vue's internal properties.

Properties that start with _ or $ will not be proxied on the Vue instance because they may conflict with Vue’s internal properties and API methods.

Try changing _value to something that doesn't start with an underscore.

Answer is from https://github.com/vuejs/vue/issues/7434

Props are read-only, but you are trying to change its value with v-model. In this case, if you change the input value, the prop is not modified and the value is restored on the next update.

Use a data property or a computed setter instead:

computed: {
  propModel: {
    get () { return this.prop },
    set (value) { this.$emit('update:prop', value) },
  },
},

https://v2.vuejs.org/v2/guide/computed.html#Computed-Setter

Bert addresses your direct issue, but I think you should also know that your approach is a bit off. Since ultimately you are sending the new value to postPost, you don't really need to modify your local copy. Use the event object that is sent to the change handler to get the current value from the input.

Instead of v-model, just use :value, and don't include the invocation parentheses when specifying the change handler.

<template>
<div>
       <b-form-input :value="value" @change="postPost"></b-form-input>
</div>
</template>
<script>
    import axios from 'axios';
    export default {
        props: {
            value: String
        },
        methods: {
            postPost(event) {
                axios.put('/trajectory/inclination', {
                    body: event.target.value
                })
                    .then(response => {
                    })
                    .catch(e => {
                        this.errors.push(e)
                    })
            }
        }
    }
</script>