I could give you a straightforward answer, but I'm guessing you - and anyone else who finds this question later - would prefer to really understand the answer. So, this is going to take a bit of background to explain, and will be a bit long. Stick with it, and hopefully it will end up relatively clear.
There are different circumstances in which you get a verb followed by the -ing form of a verb. Once you want to add an adverbial phrase or clause, or just a one-word adverb, it gets more complex as it could apply to any of the verbs involved.
I was running
In this case, we have two verbs, was and running. However, was is an auxiliary helping to form the past progressive, and running is the progressive participle, so if we add an adverb, or even an adverbial clause:
I was running slowly
I was running because I thought I was going to be late
There's no confusion as to how to parse it; the adverbial applies to running.
I remembered skating
Here we have a catenative verb in the shape of to remember. Catenative verbs take another verb or verb phrase as an object. Sometimes they take a to-infinitive, sometimes they take a bare infinitive, and sometimes they take a gerund, the other grammatical role of the -ing form of a verb. In some cases, a catenative verb can take different forms of catenated verb, and they may change meaning slightly in each case. Whichever form of verb they take, it can have its own object(s), complements, adverbs and so on:
I remembered skating with Jane
I remembered skating slowly
I remembered skating to school
But what if you want an adverb to modify the catenative verb itself? Well, it's not ambiguous if it goes first:
I dimly remembered skating
But what about these:
I remembered dimly skating
I remembered, dimly, skating
I remembered skating dimly
I remembered skating, dimly
The first example, with the adverb interposed between the catenative verb and the gerund, is ambiguous grammatically, but because it's hard to see what it might mean to skate in a dim manner, a reader would usually attach it to remembered. If the adverb were quickly they would be more likely to assume the other way.
The second example, with the adverb interposed yet separated by commas, attaches it to the first verb. "I remembered, quickly, skating" means that it was the remembering that was done quickly, not the skating.
The third is not ambiguous; the adverb quickly should, and will, be read as attaching to skating. The fourth is once again ambiguous, and the semantic dimension will play a part - as will context.
Now, come can also be catenative. It can be intransitive, as well. In either case, it might take the -ing form - it could be the catenated verb, in the same way as "would you like to come dancing with me?" or it could be a descriptive gerund, which is another situation in which a verb will be followed by the -ing form of another verb.
For example, we might say "it's good to die laughing", where the gerund describes something else being done at the same time as the verb it is applied to. "Here she comes, dancing down the street" applies the same principle with a full gerund phrase.
So, you start with the following sentence:
We had a strange woman come to the door selling pictures
Here, to the door effectively the argument of the verb to come, describing where the strange woman came. 'Selling pictures is a descriptive gerund phrase, describing something else she was doing when she came to the door.
Now, the construction 'come selling pictures' is valid, with come in its catenative sense. However, it would be used in situations like:
I have to go to work. Would you like to come selling pictures?
The selling gerund has pictures as its object, and this phrase is the object of to come. By the way, like is also catenative here, but followed by a to-infinitive. In any case, let us consider your proposed re-write:
We had a strange woman come selling pictures to the door
The thing is, we could parse this in a few ways. You want, essentially to parse it as:
(come (selling pictures)) to the door
Now, that could make sense with come in its catenative sense, but in this context that would be semantically odd. It could make sense with selling pictures as a descriptive gerund, but doing that with this word order would be very unusual. Instead, the most likely way a native speaker will parse this is:
come (selling pictures to the door)
Here, it's a little hard to say whether the gerund phrase is a catenated object or a descriptive gerund, but that doesn't matter a great deal. The overall sense is the same. This may generate cruel laughter, as doors are not known for purchasing artwork.
However, you can get the word order you want with the right sense. It just needs some punctuation:
We had a strange woman come, selling pictures, to the door
This is relatively unusual in most dialects I'm familiar with, but I'm confident that it would be generally understood.