Posting to a form via external URL
There is no special trick to posting to Drupal Forms.
But, the way that form is set up, at least for validation errors, is ajax-based. Which means that you need to post to this URL instead:

https://www.modelaircraft.org/membership/verify?ajax_form=1&amp;_wrapper_format=drupal_ajax


with the following parameters

_drupal_ajax    1
_triggering_element_name    op
_triggering_element_value   Submit
ajax_page_state[libraries]  classy/base,classy/messages,core/drupal.ajax,core/html5shiv,core/jquery.form,core/normalize,extlink/drupal.extlink,google_analytics/google_analytics,nt_ama/nt_ama,omega/omega,responsive_menu/responsive_menu.javascript,responsive_menu/responsive_menu.styling,system/base,tb_megamenu/theme.tb_megamenu
ajax_page_state[theme_token]    
ajax_page_state[theme]  nt_ama
ama_number  123
form_build_id   form-pqXufm7ga3qhXz5_dpM6_AiGcgcExovHqBG6kkqcIIE
form_id membership_verify_form
last_name   smith


That should give you a json response, similar to this:

[{
    "command": "update_build_id",
    "old": "form-pqXufm7ga3qhXz5_dpM6_AiGcgcExovHqBG6kkqcIIE",
    "new": "form-_2pgVtOL0WW10yWGP8iS1o95GyJ5pMxo5M1Mk3Jy90c"
}, {
    "command": "insert",
    "method": "html",
    "selector": "#messages-wrapper",
    "data": "    \u003Cdiv role=\u0022contentinfo\u0022 aria-label=\u0022Error message\u0022 class=\u0022messages messages--error\u0022\u003E\n          \u003Cdiv role=\u0022alert\u0022\u003E\n                  \u003Ch2 class=\u0022visually-hidden\u0022\u003EError message\u003C\/h2\u003E\n                    No match found for AMA 123 and last name smith. Please verify.\n                \u003C\/div\u003E\n      \u003C\/div\u003E\n    ",
    "settings": null
}]


which contains the message you mention.

Good luck!