Hi Maxime,

I used the below lightning component code to open a lightning component which has a visualforce page that is rendered as pdf and below this, you can have the button to download vfpage that is rendered as pdf.

The lightning component I used is below 
 

<aura:component controller='OpenModelPopup' implements="force:appHostable,flexipage:availableForAllPageTypes,flexipage:availableForRecordHome,force:hasRecordId,forceCommunity:availableForAllPageTypes,force:lightningQuickAction" access="global" >
    
    
    <aura:attribute name="openmodel" type="boolean"/>
    <aura:attribute name="selectedLookUpRecord" type="sObject" default="{}"/>
    <aura:handler name="init" value="{!this}" action="{!c.fetchListOfRecordTypes}"/>
    <aura:attribute name="lstOfRecordType" type="String[]" />
    
    <lightning:button label="Click Me" variant="neutral" onclick="{!c.PerformAction}" />
    
    <aura:If isTrue="{!v.openmodel}">
        
        <div aura:id="editDialog" role="dialog" tabindex="-1" aria-labelledby="header43" class="slds-modal slds-fade-in-open slds-modal_small slds-backdrop ">
            <div class="slds-modal__container ">
                <div class="slds-modal__header">
                    <h2 class="slds-text-heading--medium">Modal Popup with button click</h2>
                    <button class="slds-button slds-modal__close slds-button--icon" title="Close" onclick="{!c.closeModal}">
                        <lightning:icon iconName="utility:close" size="medium" />
                        <span class="slds-assistive-text">Close</span>
                    </button>
                </div>
                <div class="slds-modal__content slds-p-around--medium slds-grid slds-wrap " style="height:400px;" >
                    <!--  Write Your Content here  -->
                    <div class="slds-size--1-of-2 slds-large-size--1-of-2 ">
                        <div class="slds-text-heading_large slds-text-color_success ">
                            <c:js/>
                            <lightning:button label="Download Document" onclick="{!c.downloadDocument}" />
                            
                        </div>
                        <br/>
                    </div>
                </div>
            </div>
        </div>
    </aura:If>
</aura:component>

 

({
    PerformAction : function(component, event, helper) {
        component.set("v.openmodel",true);
    },
    closeModal:function(component,event,helper){    
        var cmpTarget = component.find('editDialog');
        var cmpBack = component.find('overlay');
        $A.util.removeClass(cmpBack,'slds-backdrop--open');
        $A.util.removeClass(cmpTarget, 'slds-fade-in-open');
        component.set('v.openmodel',false);
        
    },
    fetchListOfRecordTypes: function(component, event, helper) {
        var action = component.get("c.fetchRecordTypeValues");
        action.setCallback(this, function(response) {
            component.set("v.lstOfRecordType", response.getReturnValue());
        });
        $A.enqueueAction(action);
    },
    
    })

 

public class OpenModelPopup{
    public static Map<Id, String> recordtypemap {get;set;}
    
    @AuraEnabled        
    public static List<String> fetchRecordTypeValues(){
        List<Schema.RecordTypeInfo> recordtypes = Account.SObjectType.getDescribe().getRecordTypeInfos();    
        recordtypemap = new Map<Id, String>();
        for(RecordTypeInfo rt : recordtypes){
            if(rt.getName() != 'Master')
                recordtypemap.put(rt.getRecordTypeId(), rt.getName());
        }        
        return recordtypemap.values();
    }    
}

The code for component with visualfforce page in it is:
 

<aura:component implements="lightning:availableForFlowScreens,force:appHostable,flexipage:availableForAllPageTypes,flexipage:availableForRecordHome,force:hasRecordId,forceCommunity:availableForAllPageTypes,force:lightningQuickAction" access="global" >
	<iframe src="{! 'https://anup2-dev-ed--c.visualforce.com/apex/testtest' }" width="100%" height="500px;" frameBorder="0"/>
</aura:component>

Below is the output I got:



For the button to save the document you can use this link to refer this implementation:

>> https://www.vermanshul.com/2017/07/lightning-generate-pdf-from-lightning.html

I hope this helps and in case if this comes in handy can you please choose this as the best answer so that it can be useful for others in the future.

Regards,
Anutej